A128302 The indices of cubes (of primes) in the 3-almost primes.

1, 5, 30, 82, 328, 551, 1243, 1763, 3112, 6276, 7619, 12972, 17615, 20322, 26514, 37977, 52220, 57703, 76200, 90701, 98470, 124541, 144229, 177395, 229275, 258410, 273908, 306750, 324149, 360724, 510225, 559384, 638657, 666743, 819645, 852588

Offset

1,2

Comments

the primepi function might be used to find terms. But it is expensive for larger numbers so then one might use A335331 to ease finding primepi(m) for larger m. - David A. Corneth, Apr 13 2025

Links

Amiram Eldar, Table of n, a(n) for n = 1..590

David A. Corneth, PARI program

Formula

A014612(a(n)) = A030078(n) = A000040(n)^3.

Example

a(4) = 82 as 343 = 7^3 = prime(4)^3, the fourth cube in the 3-almost primes, is the eighty-second 3-almost prime.

Mathematica

Position[Select[Range[10^6], PrimeOmega[#] == 3 &], _?(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) == 3, c++; if(omega(f) == 1, print1(c, ", ")))); } \\ Amiram Eldar, Apr 13 2025
(PARI) \\ See Corneth link
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi, prime
def A128302(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)**3
    return int(sum(primepi(m//prod(c[1] for c in a))-a[-1][0] for a in g(m, 0, 1, 1, 3))) # Chai Wah Wu, Jan 02 2026

Crossrefs

Cf. A000040, A014612, A030078, A128301, A128303, A335331.

Sequence in context: A344070 A273480 A164015 * A258582 A288679 A071252

Adjacent sequences: A128299 A128300 A128301 * A128303 A128304 A128305

Keyword

nonn

Author

Rick L. Shepherd, Feb 25 2007

Status

approved