cp's OEIS Frontend

No other terms <= 200000. - Harvey P. Dale, Dec 16 2010

No other terms <= 1320000. - Robert G. Wilson v, Dec 18 2010

There are almost certainly no further terms.

Almost certainly there are no further terms.

Comments from Donovan Johnson on the computation of this sequence, Dec 05 2010 (Start):

The number of digits of 13^k is approximately 1.114*k, so I defined an array d() that is a little bigger than 1.114 times the maximum k value to be checked. The elements of d() each are the value of a single digit of the decimal expansion of 13^k with d(1) being the least significant digit.

It's easier to see how the program works if I start with k = 2.

For k = 1, d(2) would have been set to 1 and d(1) would have been set to 3.

k = 2:

x = 13*d(1) = 13*3 = 39

y = 39\10 = 3 (integer division)

x-y*10 = 39-30 = 9, d(1) is set to 9

x = 13*d(2)+y = 13*1+3 = 16, y is the carry from previous digit

y = 16\10 = 1

x-y*10 = 16-10 = 6, d(2) is set to 6

x = 13*d(3)+y = 13*0+1 = 1, y is the carry from previous digit

y = 1\10 = 0

x-y*10 = 1-0 = 1, d(3) is set to 1

These steps would of course be inside a loop and that loop would be inside a k loop. A pointer to the most significant digit increases usually by one and sometimes by two for each successive k value checked. The number of steps of the inner loop is the size of the pointer. A scan is done from the first element to the pointer element to get the digit sum.

(End)

No other terms < 3*10^6. - Donovan Johnson, Dec 07 2010

From Donovan Johnson, Dec 03 2010: (Start)

To generate the additional terms I used PFGW.exe to get the decimal expansion for each number of the form 167^n (n <= 50000). Then I wrote a program in powerbasic to read the pfgw.out file and get the digit sums.

The digit sum is 10 times the n value for terms a(5) to a(56). (End)

I believe that this sequence is finite. - N. J. A. Sloane, Dec 05 2010

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.