A390955 Powers k^m, m > 1, where k is an odd squarefree composite.

225, 441, 1089, 1225, 1521, 2601, 3025, 3249, 3375, 4225, 4761, 5929, 7225, 7569, 8281, 8649, 9025, 9261, 11025, 12321, 13225, 14161, 15129, 16641, 17689, 19881, 20449, 21025, 24025, 25281, 25921, 27225, 31329, 33489, 34225, 34969, 35937, 38025, 40401, 41209, 42025

Offset

1,1

Comments

Odd terms in A303606.

Proper subset of A286708, in turn a proper subset of A001694.

Proper subset of A131605, in turn a proper subset of A001597.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A024556(n)-1)*A024556(n)) = 1/2 + Sum_{k>=2} ((zeta(k)/zeta(2*k))*(4^k-2^k)/(4^k-1) - P(k) - 1) = 0.014153293835002488455..., where P(k) is the prime zeta function. - Amiram Eldar, Dec 10 2025

Example

Table of n, a(n) for select n:
 n     a(n)
--------------------------------------
 1     225 =  15^2 = 3^2 *  5^2
 2     441 =  21^2 = 3^2 *  7^2
 3    1089 =  33^2 = 3^2 * 11^2
 4    1225 =  35^2 = 5^2 *  7^2
 5    1521 =  39^2 = 3^2 * 13^2
 6    2601 =  51^2 = 3^2 * 17^2
 7    3025 =  55^2 = 5^2 * 11^2
 8    3249 =  57^2 = 3^2 * 19^2
 9    3375 =  15^3 = 3^3 *  5^3
10    4225 =  65^2 = 5^2 * 13^2
11    4761 =  69^2 = 3^2 * 23^2
19   11025 = 105^2 = 3^2 *  5^2 * 7^2

Mathematica

nn = 45000; mm = Sqrt[nn]; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[mm], And[CompositeQ[#], SquareFreeQ[#]] &]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, If[OddQ[#], Sow[#]] &[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt
from sympy import mobius, integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection
def A390955(n):
    def g(x): return int(sum(mobius(k)*(x//k**2+1>>1) for k in range(1, isqrt(x)+1, 2))-primepi(x))
    def f(x): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f, n+1, n+1) # Chai Wah Wu, Dec 10 2025

Crossrefs

Cf. A001597, A001694, A005408, A024556, A126706, A131605, A286708, A303606, A390956.

Sequence in context: A363217 A373087 A216419 * A246199 A147276 A219022

Adjacent sequences: A390952 A390953 A390954 * A390956 A390957 A390958

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 07 2025

Status

approved