A363217 Odd powerful numbers that are not powers of primes.
225, 441, 675, 1089, 1125, 1225, 1323, 1521, 2025, 2601, 3025, 3087, 3249, 3267, 3375, 3969, 4225, 4563, 4761, 5625, 5929, 6075, 6125, 7225, 7569, 7803, 8281, 8575, 8649, 9025, 9261, 9747, 9801, 10125, 11025, 11907, 11979, 12321, 13225, 13689, 14161, 14283, 15125, 15129, 16641, 16875, 17689, 18225, 19773
Offset: 1
Keywords
Examples
a(1) = 225 = 3^2 * 5^2, the smallest odd number with multiple distinct prime factors, each of which have multiplicity exceeding 1. a(2) = 441 = 3^2 * 7^2, a(3) = 675 = 3^3 * 5^2, etc.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
With[{nn = 20000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[OddQ[#], ! PrimePowerQ[#]] &] ] -
PARI
isok(k) = (k>1) && (k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 28 2023
Formula
This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 1 }.
Sum_{n>=1} 1/a(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) - 1/2 - Sum_{p prime} 1/(p*(p-1)) = (2/3) * A082695 - 1/2 - A136141 = 0.0225742... . - Amiram Eldar, May 28 2023
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