A271329 a(n) is the sum of the divisors of the n-th sphenic number (A007304).

72, 96, 144, 144, 168, 216, 192, 216, 240, 252, 288, 288, 288, 324, 360, 336, 384, 360, 336, 456, 432, 384, 432, 504, 432, 528, 480, 448, 576, 480, 504, 540, 576, 648, 576, 576, 720, 576, 744, 684, 648, 576, 640, 816, 720, 756, 720, 864, 672, 792, 768, 720

Offset: 1

Author

Colin Barker, Apr 04 2016

Keywords

• nonn

Examples

			a(1) = 72 because the divisors of A007304(1) = 30 are {1,2,3,5,6,10,15,30}, the sum of which is 72.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Wikipedia, Sphenic number

Crossrefs

Cf. A000203, A007304.

Programs

Mathematica

DivisorSigma[1,#]&/@With[{upto=500},Sort[Select[Times@@@Subsets[ Prime[ Range[ Ceiling[ upto/6]]],{3}],#<=upto&]]] (* Harvey P. Dale, May 30 2020 *)

PARI

L=List(); for(n=1, 1000, if(bigomega(n)==3 && omega(n)==3, listput(L, sum(k=1, 8, divisors(n)[k])))); Vec(L)

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, divisor_sigma
def A271329(n):
    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),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    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
    return divisor_sigma(bisection(f)) # Chai Wah Wu, Aug 30 2024

Formula

a(n) = A000203(A007304(n)). - Omar E. Pol, Dec 08 2019