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A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

%I A323174 #14 Sep 23 2020 03:21:46
%S A323174 1,1,1,2,1,0,1,4,5,-4,1,2,1,-12,-3,6,1,6,1,-2,-19,-28,1,4,14,-60,19,
%T A323174 -10,1,-12,1,10,-51,-124,-12,10,1,-252,-115,0,1,-48,1,-26,7,-508,1,8,
%U A323174 41,12,-243,-58,1,22,-64,-8,-499,-1020,1,-12,1,-2044,-17,12,-168,-120,1,-122,-1011,-54,1,18,1,-4092,26,-250,-39,-264,1,4
%N A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).
%C A323174 Zeros occur at A122111(A000396(k)), k >= 1: 6, 40, 11264, 18253611008, ...
%H A323174 Antti Karttunen, <a href="/A323174/b323174.txt">Table of n, a(n) for n = 1..12000</a>
%H A323174 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A323174 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A323174 a(n) = A033879(A122111(n)).
%F A323174 a(n) = 2*A122111(n) - A323173(n).
%t A323174 A122111[n_] := Product[Prime[Sum[If[j<i, 0, 1], {j, #}]], {i, Max[#]}]&[ Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
%t A323174 a[n_] := With[{k = A122111[n]}, 2k - DivisorSigma[1, k]];
%t A323174 Array[a, 80] (* _Jean-François Alcover_, Sep 23 2020 *)
%o A323174 (PARI)
%o A323174 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A323174 A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
%o A323174 A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }
%Y A323174 Cf. A033879, A122111, A323167, A323168.
%Y A323174 Cf. also A323244.
%K A323174 sign
%O A323174 1,4
%A A323174 _Antti Karttunen_, Jan 10 2019