A388304 Numbers that have several prime factors and all prime factors have the same exponent > 2.

216, 1000, 1296, 2744, 3375, 7776, 9261, 10000, 10648, 17576, 27000, 35937, 38416, 39304, 42875, 46656, 50625, 54872, 59319, 74088, 97336, 100000, 132651, 166375, 185193, 194481, 195112, 234256, 238328, 274625, 279936, 287496, 328509, 343000, 405224, 456533, 456976, 474552

Offset

1,1

Comments

Cubefull proper powers of composite squarefree numbers. - Michael De Vlieger, Sep 20 2025

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 21.

Index entries for sequences related to powerful numbers.

Formula

This sequence U A177492 = A303606, where U denotes disjoint union.

From Michael De Vlieger, Sep 20 2025: (Start)

This sequence is { k^m : 1 < omega(k) = bigomega(k), m > 2 } = A036966 intersect A303606.

Proper subset of A131605 which in turn is a proper subset of A001597, in turn a proper subset of A001694.

Proper subset of A286708 which in turn is a proper subset of A126706, the latter is the intersection of A013929 and A024619. (End)

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A120944(n)-1)*A120944(n)^2) = Sum_{k>=3} (zeta(k)/zeta(2*k) - P(k) - 1) = 0.00790686432108743166888..., where P(k) is the prime zeta function. - Amiram Eldar, Oct 01 2025

Example

74088 is a term because 74088 = (2*3*7)^3.

Mathematica

nn = 500000; mm = Sqrt[nn]; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[mm], And[SquareFreeQ[#], CompositeQ[#]] &]]; Union@ Reap[While[j = 3; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 3, k++; i++] ][[-1, 1]] (* Michael De Vlieger, Sep 19 2025 *)

Prog

(SageMath)
def is_A388304(n: int) -> int:
    if n == 1: return False
    factors = list(factor(n))
    exponents = [f[1] for f in factors]
    return 1 != len(factors) and 1 == len(set(exponents)) and 2 < max(exponents)
print([n for n in range(1, 500000) if is_A388304(n)])
(PARI) is_a388304(n) = if(n<216, 0, my(f=factor(n), m=vecmin(f[, 2])); #f[, 1]>1 && m>2 && vecmax(f[, 2])==m) \\ Hugo Pfoertner, Sep 19 2025
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection
def A388304(n):
    if n == 1: return 216
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x, k)[0]) for k in range(3, y))
    return bisection(f, n, n) # Chai Wah Wu, Sep 20 2025

Crossrefs

Cf. A001597, A001694, A013929, A024619, A036966, A120944, A126706, A131605, A177492, A286708, A303606, A384006.

Sequence in context: A109399 A387255 A390905 * A384006 A111029 A124581

Adjacent sequences: A388301 A388302 A388303 * A388305 A388306 A388307

Keyword

nonn

Author

Peter Luschny, Sep 19 2025

Status

approved