cp's OEIS Frontend

A359281 Numbers k such that the digit sum of 5^k is a power of 5.

%I A359281 #25 Feb 13 2023 05:54:05
%S A359281 0,1,8,208,977,1007,4938,24709,24733,24853,124274,3105928
%N A359281 Numbers k such that the digit sum of 5^k is a power of 5.
%C A359281 The number of digits in the decimal expansion of 5^k is 1 + floor(log_10(5^k)). If the average digit value is approximately (0 + 9)/2 = 9/2, then for large values of k, the digit sum will be approximately (9/2)*log_10(5^k) = (9/2)*k*log_10(5). The digit sum will then tend to be in the vicinity of a power of 5 when log_5((9/2)*k*log_10(5)) is near an integer, i.e., when log_5((9/2)*log_10(5)) + log_5(k) = 0.7120063... + log_5(k) is near an integer, which happens when k is near 5^(j - 0.7120063...) for integers j, i.e., around k = 1.59, 7.95, 39.7, 199, 993, 4968, 24838, 124191, 620953, etc. - _Jon E. Schoenfield_, Dec 24 2022
%F A359281 A067502(n) = 5^a(n).
%e A359281 5^8 = 390625 and 3+9+0+6+2+5 = 5^2, so 8 is a term.
%p A359281 filter:= proc(n) local x;
%p A359281  x:= convert(convert(5^n,base,10),`+`);
%p A359281    x = 5^padic:-ordp(x,5)
%p A359281 end proc:
%p A359281 select(filter, [$0..10^5]); # _Robert Israel_, Jan 18 2023
%t A359281 Do[If[IntegerQ[Log[5, Plus @@ IntegerDigits[5^n]]], Print[n]], {n, 0, 150000}];
%o A359281 (PARI) isok5(k) = (k==1) || (k==5) || (ispower(k,,&p) && (p==5));
%o A359281 isok(k) = isok5(sumdigits(5^k)); \\ _Michel Marcus_, Dec 24 2022
%Y A359281 Cf. A000351, A066001 (sum of digits of 5^n), A067502.
%K A359281 nonn,base,more,hard
%O A359281 1,3
%A A359281 _David Radcliffe_, Dec 23 2022
%E A359281 a(11) from _Michal Paulovic_, Jan 18 2023