A392741 Powers k^m, m > 1, squarefree k with more than 2 distinct prime factors.

900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27000, 27225, 28900, 30276, 33124, 34596, 36100, 38025, 44100, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 74088, 74529, 79524, 81225, 81796, 84100, 96100, 101124, 103684

Offset

1,1

Comments

Perfect powers of k in A350352.

This sequence is A303606 \ A303661.

Intersection of A000977 and A303606.

Superset of A162143, superset of A391320.

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_{k>=2} (zeta(k)/zeta(2*k) - P(k) - (P(k)^2 - P(2*k))/2 - 1) = 0.0038711835510255541315..., where P(k) is the prime zeta function. - Amiram Eldar, Mar 14 2026

Example

Table of n, a(n) for select n:
    n        a(n)
  ----------------------------------------------
    1      900 =  30^2 = 2^2 * 3^2 *  5^2
    2     1764 =  42^2 = 2^2 * 3^2 *  7^2
    3     4356 =  66^2 = 2^2 * 3^2 * 11^2
    4     4900 =  70^2 = 2^2 * 5^2 *  7^2
    5     6084 =  78^2 = 2^2 * 3^2 * 13^2
    6    10404 = 102^2 = 2^2 * 3^2 * 17^2
    7    11025 = 105^2 = 3^2 * 5^2 *  7^2
   13    27000 =  30^3 = 2^3 * 3^3 *  5^3
   21    44100 = 210^2 = 2^2 * 3^2 *  5^2 * 7^2
  138   810000 =  30^4 = 2^4 * 3^4 *  5^4

Mathematica

nn = 100000; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], 2 < PrimeNu[#] == PrimeOmega[#] &]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

Prog

(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection
def A392741(n):
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) for d in g(x, a2, b2, c*b2, m-1)))
    def h(x): return sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, i)) for i in range(3, x.bit_length()))
    def f(x): return int(n+x-sum(h(integer_nthroot(x, i)[0]) for i in range(2, x.bit_length())))
    return bisection(f, n, n) # Chai Wah Wu, Mar 09 2026

Crossrefs

Cf. A000977, A001597, A001694, A072777, A120944, A126706, A131605, A162143, A286708, A303606, A303661, A350352, A375055, A390950, A391320.

Sequence in context: A386798 A391427 A074853 * A391320 A162143 A391428

Adjacent sequences: A392738 A392739 A392740 * A392742 A392743 A392744

Keyword

nonn,easy

Author

Michael De Vlieger, Feb 28 2026

Status

approved