A067863 Numbers k such that k divides the sum of digits of 7^k.

1, 13, 67, 94, 139, 220, 805

Offset: 1

Author

Shyam Sunder Gupta and Amarnath Murthy, Feb 16 2002

Keywords

• hard
• more
• nonn
• base

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

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

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

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

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

The probability P(k) that s(d)/k departs from the mean by an amount at least sufficient to reach the nearest higher or lower integer (so that k 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 >= k (this sum is less than 10^-33 at k=100000). (End)

Examples

			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.
		

Crossrefs

Cf. A062927, A067862, A067864.

Programs

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 *)
Select[Range[1000],Mod[Total[IntegerDigits[7^#]],#]==0&] (* Harvey P. Dale, Sep 22 2024 *)

Extensions

Edited by Jon E. Schoenfield, May 29 2010