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

%I #9 Oct 20 2024 12:38:51

%S 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,

%T 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,

%U 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

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

%C The nonzero subsequence of A123074.

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot, primefactors

%o def A123073(n):

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

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

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o 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)))

%o return (1,3,6)[len(primefactors(bisection(f,n,n)))-1] # _Chai Wah Wu_, Oct 20 2024

%Y Cf. A123074, A014612.

%K nonn

%O 1,2

%A _N. J. A. Sloane_ and T. D. Noe, Sep 29 2006

%E More terms from _T. D. Noe_, Sep 29 2006