A123073 - OEIS

A123073

Number of ordered triples of primes (p,q,r) such that pqr = n-th 3-almost prime A014612(n).

1, 3, 3, 3, 1, 3, 6, 6, 3, 3, 3, 3, 3, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 1, 6, 6, 3, 3, 3, 6, 3, 6, 6, 3, 3, 6, 3, 6, 6, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 3, 6, 3, 3, 6, 3, 6, 6, 6, 3, 6, 3, 6, 6, 3, 3, 3, 3, 1, 6, 6, 3, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 3, 6, 3

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OFFSET

1,2

COMMENTS

The nonzero subsequence of A123074.

LINKS

Table of n, a(n) for n=1..105.

PROG

(Python)

from math import isqrt

from sympy import primepi, primerange, integer_nthroot, primefactors

def A123073(n):

def bisection(f, kmin=0, kmax=1):

while f(kmax) > kmax: kmax <<= 1

while kmax-kmin > 1:

kmid = kmax+kmin>>1

if f(kmid) <= kmid:

kmax = kmid

else:

kmin = kmid

return kmax

def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1)) for b, m in enumerate(primerange(k, isqrt(x//k)+1), a)))

return (1, 3, 6)[len(primefactors(bisection(f, n, n)))-1] # Chai Wah Wu, Oct 20 2024

CROSSREFS

Cf. A123074, A014612.

Sequence in context: A181520 A256736 A372700 * A059289 A201277 A163644

Adjacent sequences: A123070 A123071 A123072 * A123074 A123075 A123076

KEYWORD

nonn

AUTHOR

N. J. A. Sloane and T. D. Noe, Sep 29 2006

STATUS

approved