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

%I A271329 #20 Aug 31 2024 01:57:21
%S A271329 72,96,144,144,168,216,192,216,240,252,288,288,288,324,360,336,384,
%T A271329 360,336,456,432,384,432,504,432,528,480,448,576,480,504,540,576,648,
%U A271329 576,576,720,576,744,684,648,576,640,816,720,756,720,864,672,792,768,720
%N A271329 a(n) is the sum of the divisors of the n-th sphenic number (A007304).
%H A271329 Colin Barker, <a href="/A271329/b271329.txt">Table of n, a(n) for n = 1..1000</a>
%H A271329 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sphenic_number">Sphenic number</a>
%F A271329 a(n) = A000203(A007304(n)). - _Omar E. Pol_, Dec 08 2019
%e A271329 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.
%t A271329 DivisorSigma[1,#]&/@With[{upto=500},Sort[Select[Times@@@Subsets[ Prime[ Range[ Ceiling[ upto/6]]],{3}],#<=upto&]]] (* _Harvey P. Dale_, May 30 2020 *)
%o A271329 (PARI)
%o A271329 L=List(); for(n=1, 1000, if(bigomega(n)==3 && omega(n)==3, listput(L, sum(k=1, 8, divisors(n)[k])))); Vec(L)
%o A271329 (Python)
%o A271329 from math import isqrt
%o A271329 from sympy import primepi, primerange, integer_nthroot, divisor_sigma
%o A271329 def A271329(n):
%o A271329     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)))
%o A271329     def bisection(f,kmin=0,kmax=1):
%o A271329         while f(kmax) > kmax: kmax <<= 1
%o A271329         while kmax-kmin > 1:
%o A271329             kmid = kmax+kmin>>1
%o A271329             if f(kmid) <= kmid:
%o A271329                 kmax = kmid
%o A271329             else:
%o A271329                 kmin = kmid
%o A271329         return kmax
%o A271329     return divisor_sigma(bisection(f)) # _Chai Wah Wu_, Aug 30 2024
%Y A271329 Cf. A000203, A007304.
%K A271329 nonn
%O A271329 1,1
%A A271329 _Colin Barker_, Apr 04 2016