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A062822 Sum of divisors of the squarefree numbers: sigma(A005117(n)).

%I A062822 #27 Aug 13 2024 01:56:09
%S A062822 1,3,4,6,12,8,18,12,14,24,24,18,20,32,36,24,42,30,72,32,48,54,48,38,
%T A062822 60,56,42,96,44,72,48,72,54,72,80,90,60,62,96,84,144,68,96,144,72,74,
%U A062822 114,96,168,80,126,84,108,132,120,90,112,128,144,120,98,102,216,104,192
%N A062822 Sum of divisors of the squarefree numbers: sigma(A005117(n)).
%H A062822 Harvey P. Dale, <a href="/A062822/b062822.txt">Table of n, a(n) for n = 1..1000</a>
%F A062822 a(n) = Product_{k=1..A001221(n)} (A265668(n,k) + 1). - _Reinhard Zumkeller_, Dec 13 2015
%F A062822 From _Amiram Eldar_, Nov 21 2022: (Start)
%F A062822 a(n) = A000203(A005117(n)).
%F A062822 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/(72*zeta(3)) = A152649 / A002117 = 1.1254908... . (End)
%t A062822 DivisorSigma[1,#]&/@Select[Range[150],SquareFreeQ] (* _Harvey P. Dale_, May 18 2014 *)
%o A062822 (PARI) j=[]; for(n=1,200, if(issquarefree(n),j=concat(j, sigma(n)))); j
%o A062822 (Haskell)
%o A062822 a062822 1 = 1
%o A062822 a062822 n = product $ map (+ 1) $ a265668_row n
%o A062822 -- _Reinhard Zumkeller_, Dec 13 2015
%o A062822 (Python)
%o A062822 from math import isqrt
%o A062822 from sympy import mobius, divisor_sigma
%o A062822 def A062822(n):
%o A062822     def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A062822     m, k = n, f(n)
%o A062822     while m != k:
%o A062822         m, k = k, f(k)
%o A062822     return divisor_sigma(m) # _Chai Wah Wu_, Aug 12 2024
%Y A062822 Cf. A000203, A005117, A002117, A152649, A265668, A001221.
%K A062822 nonn
%O A062822 1,2
%A A062822 _Jason Earls_, Jul 20 2001