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A320779 Inverse Euler transform of the number of divisors function A000005.

%I A320779 #21 Jul 15 2024 15:33:16
%S A320779 1,1,0,0,-1,1,-1,0,1,-1,0,1,-1,-1,2,1,-2,-2,2,3,-4,0,3,-3,3,-2,-2,2,1,
%T A320779 7,-15,0,17,-11,-1,0,9,-4,-18,26,-10,-10,24,-17,-15,21,27,-42,-37,69,
%U A320779 43,-113,-11,149,-98,-24,67,-57,24,-53,213,-243,-193,704
%N A320779 Inverse Euler transform of the number of divisors function A000005.
%C A320779 The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n).
%H A320779 Alois P. Heinz, <a href="/A320779/b320779.txt">Table of n, a(n) for n = 1..5000</a>
%H A320779 OEIS Wiki, <a href="https://oeis.org/wiki/Euler_transform">Euler transform</a>
%p A320779 # The function EulerInvTransform is defined in A358451.
%p A320779 a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 0))):
%p A320779 seq(a(n), n = 1..64); # _Peter Luschny_, Nov 21 2022
%t A320779 EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
%t A320779 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
%t A320779 EulerInvTransform[Table[DivisorSigma[0,n],{n,100}]]
%o A320779 (Python)
%o A320779 from functools import lru_cache
%o A320779 from sympy import mobius, divisors, divisor_count
%o A320779 def A320779(n):
%o A320779     @lru_cache(maxsize=None)
%o A320779     def b(n): return divisor_count(n)
%o A320779     @lru_cache(maxsize=None)
%o A320779     def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
%o A320779     return sum(mobius(d)*c(n//d) for d in divisors(n,generator=True))//n # _Chai Wah Wu_, Jul 15 2024
%Y A320779 Cf. A000005.
%K A320779 sign
%O A320779 1,15
%A A320779 _Gus Wiseman_, Oct 22 2018