A128304 Indices of 4th powers (of primes) in the 4-almost primes.

1, 8, 90, 385, 2556, 5138, 15590, 24646, 53993, 139199, 182476, 375363, 569617, 691012, 991150, 1613115, 2490040, 2849478, 4163793, 5263838, 5888203, 8100907, 9886861, 13102875, 18538021, 21816421, 23608907, 27525519, 29659283, 34290671

Offset

1,2

Links

Chai Wah Wu, Table of n, a(n) for n = 1..560 (terms 1..40 from Zak Seidov, terms 41..99 from Amiram Eldar)

Formula

A014613(a(n)) = A030514(n) = A000040(n)^4.

Example

a(3) = 90 as 625 = 5^4 = prime(3)^4, the third 4th power in the 4-almost primes, is the 90th 4-almost prime.

Mathematica

Position[Select[Range[10^6], PrimeOmega[#] == 4 &], _?(PrimeNu[#] == 1 &)] // Flatten (* Amiram Eldar, Apr 13 2025 *)

Prog

(PARI) list(lim) = {my(f, c); for(k = 1, lim, f = factor(k); if(bigomega(f) == 4, c++; if(omega(f) == 1, print1(c, ", ")))); } \\ Amiram Eldar, Apr 13 2025
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi, prime
def A128304(n):
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
    m = prime(n)**4
    return int(sum(primepi(m//prod(c[1] for c in a))-a[-1][0] for a in g(m, 0, 1, 1, 4))) # Chai Wah Wu, Jan 02 2026

Crossrefs

Cf. A000040, A014613, A030514, A128303, A128305.

Sequence in context: A045732 A371208 A077192 * A247728 A056784 A166769

Adjacent sequences: A128301 A128302 A128303 * A128305 A128306 A128307

Keyword

nonn

Author

Rick L. Shepherd, Mar 05 2007

Status

approved