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%I #20 May 26 2018 21:58:10
%S 1,3,6,11,17,25,34,45,57,71,86,103,121,141,162,184,208,233,260,288,
%T 318,349,382,416,452,489,528,568,610,653,697,743,790,839,889,941,994,
%U 1049,1105,1163,1222,1283,1345,1408,1473,1539,1607,1676,1747,1819,1893,1968
%N Index of 5^n within sequence of numbers of form 3^i*5^j.
%C Write down the numbers 3^i * 5^j in an ordered list and then record where the powers of 5 appear.
%H Michael De Vlieger, <a href="/A022338/b022338.txt">Table of n, a(n) for n = 1..1000</a>
%F From _David A. Corneth_, May 14 2018: (Start)
%F Numbers between 5^n and 5^(n + 1) are of the form 5^m * 3^j where j > 0 and so m < n.
%F Thus 5^n < 5^m * 3^j < 5^(n + 1) if and only if 5^(n - m) < 3^j < 5^(n - m + 1).
%F Taking logs give (n - m) * log(5) < j * log(3) < (n - m + 1) * log(5).
%F Dividing by log(3) > 0 gives (n - m) * log(5) / log(3) < j < (n - m + 1) * log(5) / log(3).
%F (End)
%e 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.
%e 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).
%t 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 *)
%Y Cf. A000351 (powers of 5), A003593 (odd 5-smooth numbers), A025715.
%K nonn
%O 1,2
%A _Clark Kimberling_
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