cp's OEIS Frontend

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)

a(36) >= 423378.

Please consult the argument in A067863 for the reason that it is believed that all individual such sequences (all k's which divide b^k) terminate.