A226724 Positions of the numbers 5^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

5, 9, 14, 19, 24, 28, 34, 38, 43, 48, 53, 57, 63, 67, 71, 77, 81, 86, 91, 96, 100, 106, 110, 115, 120, 125, 129, 135, 139, 143, 148, 153, 158, 162, 168, 172, 177, 182, 187, 191, 197, 201, 205, 211, 215, 220, 225, 230, 234, 240, 244, 249, 254, 259, 263, 269

Offset

1,1

Links

Clark Kimberling, Table of n, a(n) for n = 1..2000

Formula

a(n) = 1 + n + floor(n*log_2(5)) + floor(n*log_3(5)).

Example

The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 5^n for n >= 0 are in positions 5, 9, 14.

Mathematica

z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

Prog

(Python)
from sympy import integer_log
def A226724(n): return n+(m:=5**n).bit_length()+integer_log(m, 3)[0] # Chai Wah Wu, Feb 03 2026

Crossrefs

Cf. A123384, A226720, A226722, A226723, A306044.

Sequence in context: A314892 A314893 A314894 * A314895 A314896 A314897

Adjacent sequences: A226721 A226722 A226723 * A226725 A226726 A226727

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2013

Status

approved