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

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Wikipedia, Sphenic number

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A000203, A007304.

Sequence in context: A030025 A245213 A270308 * A307867 A099943 A118218

Adjacent sequences: A271326 A271327 A271328 * A271330 A271331 A271332

Keyword

nonn

Author

Colin Barker, Apr 04 2016

Status

approved