A128303 Indices of squares (of semiprimes) in the 4-almost primes.

1, 3, 8, 12, 24, 29, 59, 66, 90, 97, 162, 172, 187, 224, 234, 335, 385, 412, 489, 531, 551, 630, 692, 791, 921, 997, 1128, 1223, 1256, 1285, 1420, 1484, 1518, 1549, 1937, 2146, 2315, 2441, 2483, 2556, 2606, 2651, 2915, 3124, 3175, 3542, 3587, 3645, 3751, 3800

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

A014613(a(n)) = A074985(n) = A001358(n)^2.

Example

a(5) = 24 as 196 = 14^2 = semiprime(5)^2, the 5th square in the 4-almost primes, is the 24th 4-almost prime.

Mathematica

Position[Select[Range[3*10^4], PrimeOmega[#] == 4 &], _?(IntegerQ[Sqrt[#]] &)] // 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(vecprod(f[, 2]) == 4, print1(c, ", ")))); } \\ Amiram Eldar, Apr 13 2025
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
from oeis_sequences.OEISsequences import bisection
def A128303(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)))
    def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    m = bisection(f, n, n)**2
    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. A014613, A074985, A001358, A128302, A128304.

Sequence in context: A209934 A007434 A379717 * A123906 A395181 A065970

Adjacent sequences: A128300 A128301 A128302 * A128304 A128305 A128306

Keyword

nonn

Author

Rick L. Shepherd, Mar 04 2007

Status

approved