A025721 - OEIS

%I #26 Nov 13 2025 09:38:55

%S 1,3,7,13,21,30,41,54,69,85,103,123,145,169,194,221,250,281,313,347,

%T 383,421,460,501,544,589,636,684,734,786,840,895,952,1011,1072,1134,

%U 1198,1264,1332,1402,1473,1546,1621,1698,1776,1856,1938,2022,2108,2195,2284

%N Index of 7^n within the sequence of the numbers of the form 3^i*7^j.

%C Positions of zeros in A025642. - _R. J. Mathar_, Jul 06 2025

%H Robert Israel, <a href="/A025721/b025721.txt">Table of n, a(n) for n = 0..10000</a>

%F From _Amiram Eldar_, Nov 13 2025: (Start)

%F A003594(a(n)) = 7^n.

%F a(n) = 1 + Sum_{k=0..n} ceiling(k * c), where c = log_3(7) (A152565).

%F a(n) ~ c * n^2 / 2, where c is defined above. (End)

%e a(0) = 1, a(1) = 3 and a(2) = 7 because the first 7 numbers of the form 3^i * 7^j with i >= 0 and j >= 0 are 1, 3, 7, 9, 21, 27 and 49. - _Robert Israel_, Aug 20 2024

%p dA:= map(t -> 1+ilog[3](7^t), [$0..100]):

%p ListTools:-PartialSums(dA); # _Robert Israel_, Aug 20 2024

%t Accumulate[Table[Ceiling[n * Log[3, 7]], {n, 0, 60}]] + 1 (* _Amiram Eldar_, Nov 13 2025 *)

%Y Cf. A000420, A003594, A025642, A152565.

%K nonn,easy

%O 0,2

%A _David W. Wilson_

%E Offset corrected by _Amiram Eldar_, Nov 13 2025