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A067863 Numbers n such that n divides the sum of digits of 7^n. 7
1, 13, 67, 94, 139, 220, 805 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

There are no other terms < 3000. - Stefan Steinerberger, Mar 14 2006

No more terms < 50000. - David Wasserman, May 30 2008

From Jon E. Schoenfield, May 29 2010: (Start)

No more terms < 100000. It is nearly certain that no terms exist beyond 805.

Let f(n) be the sum of digits of 7^n. Let d be the number of digits, i.e., d=ceiling(log_10(7^n)).

Let s(k) be the sum of k random digits (each drawn independently from a uniform distribution over the integers 0 through 9).

As n increases, the behavior of f(n)/n becomes increasingly similar to that of s(d)/n.

The mean and variance of s(d)/n are 4.5*d/n and 28.5*d/n^2, respectively.

For large values of n, the distribution of s(d)/n approaches a standard normal distribution with mean 4.5*log_10(7) (approximately 3.80294) and variance 28.5*log_10(7)/n.

The probability P(n) that s(d)/n departs from the mean by an amount at least sufficient to reach the nearest higher or lower integer (so that n divides the sum of digits) becomes vanishingly small (e.g., P(50000) < 10^-18, P(100000) < 10^-36, P(150000) < 10^-54), and the same is true of the sum of P(i) for all i>=n (this sum is less than 10^-33 at n=100000). (End)

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

13 divides the sum of digits of 7^13 (i.e., 9 + 6 + 8 + 8 + 9 + 0 + 1 + 0 + 4 + 0 + 7 = 52), so 13 is in the sequence.

MATHEMATICA

For[n = 1, n < 2000, n++, a := DigitCount[7^n]; If[IntegerQ[Sum[a[[i]]*i, {i, 1, 9}]/n], Print[n]]] (* Stefan Steinerberger, Mar 14 2006 *)

CROSSREFS

Cf. A062927, A067862, A067864.

Sequence in context: A041320 A058380 A129746 * A257809 A106975 A086689

Adjacent sequences:  A067860 A067861 A067862 * A067864 A067865 A067866

KEYWORD

hard,more,nonn,base

AUTHOR

Shyam Sunder Gupta and Amarnath Murthy, Feb 16 2002

EXTENSIONS

Edited by Jon E. Schoenfield, May 29 2010

STATUS

approved

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Last modified January 16 03:30 EST 2019. Contains 319184 sequences. (Running on oeis4.)