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Автор: Grafiati

Опубліковано: 8 червня 2024

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1

Costa, Eudes Antonio, Deyfila Da Silva Lima, Élis Gardel da Costa Mesquita, and Keidna Cristiane Oliveira Souza. "Soma iterada de algarismos de um número racional." Ciência e Natura 43 (March 1, 2021): e12. http://dx.doi.org/10.5902/2179460x41972.

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The digital roots S* (x), of a n positive integer is the digit 0 ≤ b ≤ 9 obtained through an iterative digit sum process, where each iteration is obtained from the previous result so that only the b digit remains. For example, the iterated sum of 999999 is 9 because 9 + 9 + 9 + 9 + 9 + 9 = 54 and 5 + 4 = 9. The sum of the digits of a positive integer, and even the digital roots, is a recurring subject in mathematical competitions and has been addressed in several papers, for example in Ghannam (2012), Ismirli (2014) or Lin (2016). Here we extend the application Sast to a positive rational number x with finite decimal representation. We highlight the following result: given a rational number x, with finite decimal representation, and the sum of its digits is 9, so when divided x by powers of 2, the number resulting also has the sum of its digits 9. Fact that also occurs when the x number is divided by powers of 5. Similar results were found when the x digit sum is 3 or 6.

2

Mooney, Edward S. "Solve It!: A Mix of 1, 2, 3, 4, 5, 6." Mathematics Teaching in the Middle School 13, no. 4 (November 2007): 218–19. http://dx.doi.org/10.5951/mtms.13.4.0218.

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3

KESSEBÖHMER, MARC, and MEHDI SLASSI. "LARGE DEVIATION ASYMPTOTICS FOR CONTINUED FRACTION EXPANSIONS." Stochastics and Dynamics 08, no. 01 (March 2008): 103–13. http://dx.doi.org/10.1142/s0219493708002226.

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We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation processes. Also a large deviation asymptotic for single digits is given.

4

Allaart, Pieter C. "An invariant-sum characterization of Benford's law." Journal of Applied Probability 34, no. 1 (March 1997): 288–91. http://dx.doi.org/10.2307/3215195.

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The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk is constant for every fixed k.

5

Allaart, Pieter C. "An invariant-sum characterization of Benford's law." Journal of Applied Probability 34, no. 01 (March 1997): 288–91. http://dx.doi.org/10.1017/s0021900200100907.

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The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk is constant for every fixed k.

6

Hoslar, Alyssa M. "Math for Real: How to Check Digits." Mathematics Teaching in the Middle School 16, no. 2 (September 2010): 128. http://dx.doi.org/10.5951/mtms.16.2.0128.

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When an account or product number is typed into a computer, there is often a quick way to check that the number had been entered correctly. One way is with a check digit, which is an extra digit at the end of the account number, representing the ones digit of the sum of the account digits, or some other formula.

7

Cusick, Thomas W., and Lavinia Corina Ciungu. "Sum of digits sequences modulo m." Theoretical Computer Science 412, no. 35 (August 2011): 4738–41. http://dx.doi.org/10.1016/j.tcs.2011.05.030.

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8

Drmota, Michael, Christian Mauduit, and Joël Rivat. "Primes with an average sum of digits." Compositio Mathematica 145, no. 2 (March 2009): 271–92. http://dx.doi.org/10.1112/s0010437x08003898.

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AbstractThe main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form $\sum _{p\le x} e(\alpha s_q(p))$ (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

9

Drmota, Michael, and Joël Rivat. "The Sum-of-Digits Function of Squares." Journal of the London Mathematical Society 72, no. 2 (October 2005): 273–92. http://dx.doi.org/10.1112/s0024610705006769.

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10

Morgenbesser, Johannes F. "The sum of digits of Gaussian primes." Ramanujan Journal 27, no. 1 (May 26, 2011): 43–70. http://dx.doi.org/10.1007/s11139-010-9289-3.

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11

Brown, Gavin, and John H. Williamson. "Coin tossing and sum sets." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 2 (October 1987): 211–19. http://dx.doi.org/10.1017/s1446788700029347.

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AbstractWe consider the distribution μ of numbers whose binary digits are generated from infinitely many tosses of a biased coin. It is shown that, if E has positive μ measure, then some n-fold sum of E with itself must contain an interval. This contrasts with the known result that all convolution powers of μ are singular.

12

Larcher, Gerhard, and Friedrich Pillichshammer. "Moments of the Weighted Sum-of-Digits Function." Quaestiones Mathematicae 28, no. 3 (September 2005): 321–36. http://dx.doi.org/10.2989/16073600509486132.

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13

Spiegelhofer, Lukas, and Thomas Stoll. "The sum-of-digits function on arithmetic progressions." Moscow Journal of Combinatorics and Number Theory 9, no. 1 (February 20, 2020): 43–49. http://dx.doi.org/10.2140/moscow.2020.9.43.

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14

Spiegelhofer, Lukas. "Pseudorandomness of the Ostrowski sum-of-digits function." Journal de Théorie des Nombres de Bordeaux 30, no. 2 (2018): 637–49. http://dx.doi.org/10.5802/jtnb.1042.

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15

Drmota, Michael, and Johannes Gajdosik. "The distribution of the sum-of-digits function." Journal de Théorie des Nombres de Bordeaux 10, no. 1 (1998): 17–32. http://dx.doi.org/10.5802/jtnb.216.

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16

Thuswaldner, Jörg M. "The complex sum of digits function and primes." Journal de Théorie des Nombres de Bordeaux 12, no. 1 (2000): 133–46. http://dx.doi.org/10.5802/jtnb.271.

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17

Dartyge, Cécile, and András Sárközy. "The sum of digits function in finite fields." Proceedings of the American Mathematical Society 141, no. 12 (August 8, 2013): 4119–24. http://dx.doi.org/10.1090/s0002-9939-2013-11801-0.

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18

Grabner, Peter J., Peter Kirschenhofer, and Helmut Prodinger. "The Sum-of-Digits Function for Complex Bases." Journal of the London Mathematical Society 57, no. 1 (February 1998): 20–40. http://dx.doi.org/10.1112/s0024610798005663.

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19

Drmota, Michael, Christian Mauduit, and Joël Rivat. "The sum-of-digits function of polynomial sequences." Journal of the London Mathematical Society 84, no. 1 (May 11, 2011): 81–102. http://dx.doi.org/10.1112/jlms/jdr003.

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20

Morgenbesser, Johannes F. "The sum of digits of \lfloor nc\rfloor." Acta Arithmetica 148, no. 4 (2011): 367–93. http://dx.doi.org/10.4064/aa148-4-4.

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21

Thuswaldner, Jörg M. "The Sum of Digits Function In Number Fields." Bulletin of the London Mathematical Society 30, no. 1 (January 1998): 37–45. http://dx.doi.org/10.1112/s0024609397003731.

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22

Sanna, Carlo. "On the sum of digits of the factorial." Journal of Number Theory 147 (February 2015): 836–41. http://dx.doi.org/10.1016/j.jnt.2014.09.003.

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23

HARE, KEVIN G., SHANTA LAISHRAM, and THOMAS STOLL. "THE SUM OF DIGITS OF n AND n2." International Journal of Number Theory 07, no. 07 (November 2011): 1737–52. http://dx.doi.org/10.1142/s1793042111004319.

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Анотація:

Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s2(n) = s2(n2). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case p(n) = n2, looking at the subsets of n where sq(n) = sq(n2) = k for fixed k.

24

Sidorov, N. "Sum-of-digits function for certain nonstationary-bases." Journal of Mathematical Sciences 96, no. 5 (October 1999): 3609–15. http://dx.doi.org/10.1007/bf02175837.

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25

Mišík, Ladislav, Štefan Porubský, and Oto Strauch. "Uniform Distribution of the Weighted Sum-of-Digits Functions." Uniform distribution theory 16, no. 1 (June 1, 2021): 93–126. http://dx.doi.org/10.2478/udt-2021-0005.

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Abstract The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),s q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

26

Aloui, Karam, and Firas Feki. "On the distribution of integers with missing digits under hereditary sum of digits function." Publicationes Mathematicae Debrecen 94, no. 3-4 (April 1, 2019): 337–58. http://dx.doi.org/10.5486/pmd.2019.8295.

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27

Brown, Ethan C., Michèle M. M. Mazzocco, Luke F. Rinne, and Noah S. Scanlon. "Uncanny sums and products may prompt “wise choices”: Semantic misalignment and numerical judgments." Journal of Numerical Cognition 2, no. 2 (August 5, 2016): 116–39. http://dx.doi.org/10.5964/jnc.v2i2.21.

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Automatized arithmetic can interfere with numerical judgments, and semantic misalignment may diminish this interference. We gave 92 adults two numerical priming tasks that involved semantic misalignment. We found that misalignment either facilitated or reversed arithmetic interference effects, depending on misalignment type. On our number matching task, digit pairs (as primes for sums) appeared with nouns that were either categorically aligned and concrete (e.g., pigs, goats), categorically misaligned and concrete (e.g., eels, webs), or categorically misaligned concrete and intangible (e.g., goats, tactics). Next, participants were asked whether a target digit matched either member of the previously presented digit pair. Participants were slower to reject sum vs. neutral targets on aligned/concrete and misaligned/concrete trials, but unexpectedly slower to reject neutral versus sum targets on misaligned/concrete-intangible trials. Our sentence verification task also elicited unexpected facilitation effects. Participants read a cue sentence that contained two digits, then evaluated whether a subsequent target statement was true or false. When target statements included the product of the two preceding digits, this inhibited accepting correct targets and facilitated rejecting incorrect targets, although only when semantic context did not support arithmetic. These novel findings identify a potentially facilitative role of arithmetic in semantically misaligned contexts and highlight the complex role of contextual factors in numerical processing.

28

Mauduit, Christian, Joël Rivat, and András Sárközy. "On the Digits of Sumsets." Canadian Journal of Mathematics 69, no. 3 (June 1, 2017): 595–612. http://dx.doi.org/10.4153/cjm-2016-007-2.

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29

Okie, W. R., and E. G. Okie. "Check Digits for Detecting Recording Errors in Horticultural Research: Theory and Examples." HortScience 40, no. 7 (December 2005): 1956–62. http://dx.doi.org/10.21273/hortsci.40.7.1956.

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Check digit technology is frequently used in commercial applications such as shipping labels and credit cards to flag errors in numbers as they are used. Most systems use modular arithmetic to calculate a check digit from the digits in the identification number. Check digits are little used in horticultural research because the guidelines for implementing them are neither well known nor readily accessible. The USDA–ARS stone fruit breeding program at Byron, Ga., plants thousands of trees annually, which are identified using a 2-digit year prefix followed by a sequential number that identifies the tree location in the rows. Various records are taken over the life of the tree including bloom and fruit characteristics. Selected trees are propagated and tested further. To improve the accuracy of our records we have implemented a system which uses a check number which is calculated from the identification number and then converted to a letter that is added onto the end of the identification number. The check letter is calculated by summing the products of each of the digits in the number multiplied by sequential integers, dividing this sum by 23, and converting the remainder into a letter. Adding a single letter suffix is a small change and does not add much complexity to existing data collection. The types of errors caught by this system are discussed, along with those caught by other common check digit systems. Check digit terminology and theory are also covered.

30

Morgenbesser, Johannes F. "The sum of digits of squares in Z[i]." Journal of Number Theory 130, no. 7 (July 2010): 1433–69. http://dx.doi.org/10.1016/j.jnt.2010.02.011.

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31

Schlickewei, Hans Peter. "Linear equations in integers with bounded sum of digits." Journal of Number Theory 35, no. 3 (July 1990): 335–44. http://dx.doi.org/10.1016/0022-314x(90)90121-7.

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32

Drmota, Michael, Clemens Müllner, and Lukas Spiegelhofer. "Möbius orthogonality for the Zeckendorf sum-of-digits function." Proceedings of the American Mathematical Society 146, no. 9 (May 24, 2018): 3679–91. http://dx.doi.org/10.1090/proc/14015.

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33

Drmota, Michael, and Mariusz Skałba. "The parity of the Zeckendorf sum-of-digits function." manuscripta mathematica 101, no. 3 (March 1, 2000): 361–83. http://dx.doi.org/10.1007/s002290050221.

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34

Vignat, Christophe, and Tanay Wakhare. "Finite generating functions for the sum-of-digits sequence." Ramanujan Journal 50, no. 3 (November 10, 2018): 639–84. http://dx.doi.org/10.1007/s11139-018-0065-0.

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35

Dumont, Jean Marie, and Alain Thomas. "Gaussian Asymptotic Properties of the Sum-of-Digits Function." Journal of Number Theory 62, no. 1 (January 1997): 19–38. http://dx.doi.org/10.1006/jnth.1997.2044.

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36

Zhang, Chun Hua. "The Criterion for Expressing Errors with Significant Digit." Applied Mechanics and Materials 143-144 (December 2011): 815–18. http://dx.doi.org/10.4028/www.scientific.net/amm.143-144.815.

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This paper, based on the tiny error rule (TER) in algebraic sum method, derived the criterion for correctly expressing errors with significant digit, for the first time. Then, according to the criterion, an important conclusion is further reached, namely, any error only needs to keep three significant digits at most, or it is meaningless. Thus, this paper further perfected the theory of error, and so, it is of great significance for accurately processing all measurement data.

37

Bérczes, Attila, and Florian Luca. "On the Sum of Digits of Numerators of Bernoulli Numbers." Canadian Mathematical Bulletin 56, no. 4 (December 1, 2013): 723–28. http://dx.doi.org/10.4153/cmb-2011-194-4.

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Abstract.Let b > 1 be an integer. We prove that for almost all n, the sum of the digits in base b of the numerator of the Bernoulli number B2n exceeds c log n, where c := c(b) > 0 is some constant depending on b.

38

Hare, Kevin G., Shanta Laishram, and Thomas Stoll. "Stolarsky’s conjecture and the sum of digits of polynomial values." Proceedings of the American Mathematical Society 139, no. 01 (January 1, 2011): 39. http://dx.doi.org/10.1090/s0002-9939-2010-10591-9.

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39

Stoll, Thomas. "The sum of digits of polynomial values in arithmetic progressions." Functiones et Approximatio Commentarii Mathematici 47, no. 2 (December 2012): 233–39. http://dx.doi.org/10.7169/facm/2012.47.2.7.

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40

Gupta, Shyam Sunder. "88.31 Sum of the factorials of the digits of integers." Mathematical Gazette 88, no. 512 (July 2004): 258–61. http://dx.doi.org/10.1017/s0025557200174996.

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41

Foster, D. M. E. "Averaging the sum of digits function to an even base." Proceedings of the Edinburgh Mathematical Society 35, no. 3 (October 1992): 449–55. http://dx.doi.org/10.1017/s0013091500005733.

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For a fixed integer q≧2, every positive integer where each ar(q, k) ∈ {0, 1, 2, …, q–1}. The sum of digits function α(q, k) = behaves rather erratically but on averaging has a uniform behaviour. In particular if A(q, n) = , where n > 1, then it is well known that A(q, n)∼½ ((q – 1)/log q) n log n as n→∞. For even values of q, a lower bound is now given for the difference ½S(q, n) = A(q, n)–½(q–1)[logn/logq] n, where [log n/log q] denotes the greatest integer ≦ log n/log q, complementing an earlier result for odd values of q.

42

Grabner, Peter J., та Robert F. Tichy. "α-expansions, linear recurrences, and the sum-of-digits function". Manuscripta Mathematica 70, № 1 (грудень 1991): 311–24. http://dx.doi.org/10.1007/bf02568381.

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43

Drmota, Michael, Joël Rivat, and Thomas Stoll. "The sum of digits of primes in $${\mathbb{Z}}$$ [i]." Monatshefte für Mathematik 155, no. 3-4 (August 1, 2008): 317–47. http://dx.doi.org/10.1007/s00605-008-0010-1.

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44

Drmota, Michael, and Gerhard Larcher. "The Sum-of-Digits-Function and Uniform Distribution Modulo 1." Journal of Number Theory 89, no. 1 (July 2001): 65–96. http://dx.doi.org/10.1006/jnth.2000.2628.

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45

Gittenberger, Bernhard, and Jörg M. Thuswaldner. "The Moments of the Sum-Of-Digits Function in Number Fields." Canadian Mathematical Bulletin 42, no. 1 (March 1, 1999): 68–77. http://dx.doi.org/10.4153/cmb-1999-008-0.

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AbstractWe consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange’s method we obtain the main term and smaller order terms which contain periodic fluctuations.

46

Duman, Merve, Refik Keskin, and Leman Hocaoğlu. "Padovan Numbers as Sum of Two Repdigits." Proceedings of the Bulgarian Academy of Sciences 76, no. 9 (October 1, 2023): 1326–34. http://dx.doi.org/10.7546/crabs.2023.09.02.

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Padovan sequence $$(P_{n})$$ is given by $$P_{n}=P_{n-2}+P_{n-3}$$ for $$n\geq3$$ with initial condition $$(P_{0},P_{1},P_{2})=(1,1,1)$$. A positive integer is called a repdigit if all of its digits are equal. In this study, we examine the terms of the Padovan sequence, which are the sum of two repdigits. It is shown that the largest term of the Padovan sequence which can be written as a sum of two repdigits is $$P_{18}=114=111+3.$$

47

TAN, XIAOYAN, and KANGJIE HE. "A NOTE ON THE RELATIVE GROWTH RATE OF THE MAXIMAL DIGITS IN LÜROTH EXPANSIONS." Fractals 28, no. 06 (September 2020): 2050116. http://dx.doi.org/10.1142/s0218348x20501169.

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This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.

48

Erdenebat, Erdenebileg, and Ka Lun Wong. "The error term of the sum of digital sum functions in arbitrary bases." Notes on Number Theory and Discrete Mathematics 30, no. 2 (May 19, 2024): 311–18. http://dx.doi.org/10.7546/nntdm.2024.30.2.311-318.

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Let $k$ be a non-negative integer and $q > 1$ be a positive integer. Let $s_q(k)$ be the sum of digits of $k$ written in base $q.$ In 1940, Bush proved that $A_q(x)=\sum_{k \leq x} s_q (k)$ is asymptotic to $\frac{q-1}{2}x \log_q x.$ In 1968, Trollope proved an explicit formula for the error term of $A_2(n-1),$ labeled by $-E_2(n),$ where $n$ is a positive integer. In 1975, Delange extended Trollope's result to an arbitrary base $q$ by another method and labeled the error term $nF_q(\log_q n).$ When $q=2,$ the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope's method to extend $-E_2(n)$ to $-E_q(n)$ with a general base $q,$ and we also proved directly that $nF_q(\log_q n)$ and $-E_q(n)$ are equal for any $q.$

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Pfeiffer, Oliver, and Jörg M. Thuswaldner. "Waring's Problem Restricted by a System of Sum of Digits Congruences." Quaestiones Mathematicae 30, no. 4 (December 2007): 513–23. http://dx.doi.org/10.2989/16073600709486218.

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50

Grabner, Peter, and Pierre Liardet. "Harmonic properties of the sum-of-digits function for complex bases." Acta Arithmetica 91, no. 4 (1999): 329–49. http://dx.doi.org/10.4064/aa-91-4-329-349.

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