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A363217
Odd powerful numbers that are not powers of primes.
3
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
1,1
COMMENTS
This sequence is { A286708 INTERSECT A005408 } = { A001694 INTERSECT A360769 }.
Subset of A001694, A062739, A126706, and A360769.
LINKS
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
EXAMPLE
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.
MATHEMATICA
With[{nn = 20000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[OddQ[#], ! PrimePowerQ[#]] &] ]
PROG
(PARI) isok(k) = (k>1) && (k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 28 2023
KEYWORD
nonn
AUTHOR
Michael De Vlieger, May 21 2023
STATUS
approved