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A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

%I A180114 #29 Sep 21 2024 14:48:58
%S A180114 1,7,15,13,31,31,40,63,91,57,127,195,121,217,280,133,156,255,403,183,
%T A180114 399,465,600,403,364,511,819,307,847,400,381,855,961,1240,741,931,
%U A180114 1092,1023,553,1651,781,1815,1240,1281,1093,1767,1953,871,2520,2821,993,1995
%N A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).
%H A180114 Amiram Eldar, <a href="/A180114/b180114.txt">Table of n, a(n) for n = 1..10000</a>
%H A180114 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (7).
%F A180114 From _Amiram Eldar_, May 12 2023: (Start)
%F A180114 Sum_{A001694(k) < x} a(k) = c * x^(3/2) + O(x^(23/18 + eps)), where c = A362984 * A090699 / 3 = 1.5572721108... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]
%F A180114 Sum_{k=1..n} a(k) ~ c * n^3, where c = A362984 / (3 * A090699^2) = 0.151716514097... . (End)
%e A180114 Sigma(2^2) = 7, sigma(2^3) = 15, sigma(3^2) = 13.
%p A180114 emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=sigma(n); if emin(n)>1 then L:=[op(L),sn]; print(n,ifactor(n),sn,ifactor(sn)) fi; od; od;
%t A180114 pwfQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; DivisorSigma[1, Select[ Range@ 1000, pwfQ]] (* _Giovanni Resta_, Feb 06 2018 *)
%o A180114 (PARI) lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, print1(sigma(k), ", "))); \\ _Amiram Eldar_, May 12 2023
%o A180114 (Python)
%o A180114 from itertools import count, islice
%o A180114 from math import prod
%o A180114 from sympy import factorint
%o A180114 def A180114_gen(): # generator of terms
%o A180114     for n in count(1):
%o A180114         f = factorint(n)
%o A180114         if all(e>1 for e in f.values()):
%o A180114             yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
%o A180114 A180114_list = list(islice(A180114_gen(),20)) # _Chai Wah Wu_, May 21 2023
%o A180114 (Python)
%o A180114 from math import isqrt
%o A180114 from sympy import mobius, integer_nthroot, divisor_sigma
%o A180114 def A180114(n):
%o A180114     def squarefreepi(n):
%o A180114         return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A180114     def bisection(f, kmin=0, kmax=1):
%o A180114         while f(kmax) > kmax: kmax <<= 1
%o A180114         while kmax-kmin > 1:
%o A180114             kmid = kmax+kmin>>1
%o A180114             if f(kmid) <= kmid:
%o A180114                 kmax = kmid
%o A180114             else:
%o A180114                 kmin = kmid
%o A180114         return kmax
%o A180114     def f(x):
%o A180114         c, l = n+x, 0
%o A180114         j = isqrt(x)
%o A180114         while j>1:
%o A180114             k2 = integer_nthroot(x//j**2, 3)[0]+1
%o A180114             w = squarefreepi(k2-1)
%o A180114             c -= j*(w-l)
%o A180114             l, j = w, isqrt(x//k2**3)
%o A180114         c -= squarefreepi(integer_nthroot(x, 3)[0])-l
%o A180114         return c
%o A180114     return divisor_sigma(bisection(f, n, n)) # _Chai Wah Wu_, Sep 10 2024
%Y A180114 Cf. A001694, A000203, A090699, A180090, A180097, A180098, A362984.
%K A180114 easy,nonn
%O A180114 1,2
%A A180114 _Walter Kehowski_, Aug 10 2010
%E A180114 a(1)=1 prepended by and more terms from _Giovanni Resta_, Feb 06 2018