A022338 Index of 5^n within sequence of numbers of form 3^i*5^j.

1, 3, 6, 11, 17, 25, 34, 45, 57, 71, 86, 103, 121, 141, 162, 184, 208, 233, 260, 288, 318, 349, 382, 416, 452, 489, 528, 568, 610, 653, 697, 743, 790, 839, 889, 941, 994, 1049, 1105, 1163, 1222, 1283, 1345, 1408, 1473, 1539, 1607, 1676, 1747, 1819, 1893, 1968

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Write down the numbers 3^i * 5^j in an ordered list and then record where the powers of 5 appear.

Examples

			The first twenty odd 5-smooth numbers are 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243, 375, 405, 625, 675, 729, 1125.
In that subset, the powers of 5 occur at positions 1 (corresponding to 1), 3 (corresponding to 5), 6 (corresponding to 25), 11 (corresponding to 125) and 17 (corresponding to 625).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000351 (powers of 5), A003593 (odd 5-smooth numbers), A025715.

Programs

Mathematica

Position[Sort@ Flatten@ Table[3^i * 5^j, {i, 0, Log[3, #]}, {j, 0, Log[5, #/(3^i)]}] &[15^31], ?(IntegerQ@ Log[5, #] &)][[All, 1]] (* _Michael De Vlieger, May 22 2018 *)

Formula

From David A. Corneth, May 14 2018: (Start)

Numbers between 5^n and 5^(n + 1) are of the form 5^m * 3^j where j > 0 and so m < n.

Thus 5^n < 5^m * 3^j < 5^(n + 1) if and only if 5^(n - m) < 3^j < 5^(n - m + 1).

Taking logs give (n - m) * log(5) < j * log(3) < (n - m + 1) * log(5).

Dividing by log(3) > 0 gives (n - m) * log(5) / log(3) < j < (n - m + 1) * log(5) / log(3).

(End)