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A323173 Sum of divisors computed for conjugated prime factorization: a(n) = A000203(A122111(n)).

%I A323173 #14 Sep 23 2020 03:22:00
%S A323173 1,3,7,4,15,12,31,6,13,28,63,18,127,60,39,8,255,24,511,42,91,124,1023,
%T A323173 24,40,252,31,90,2047,72,4095,12,195,508,120,32,8191,1020,403,56,
%U A323173 16383,168,32767,186,93,2044,65535,36,121,78,819,378,131071,48,280,120,1651,4092,262143,96,524287,8188,217,14,600,360,1048575,762,3315,234
%N A323173 Sum of divisors computed for conjugated prime factorization: a(n) = A000203(A122111(n)).
%H A323173 Antti Karttunen, <a href="/A323173/b323173.txt">Table of n, a(n) for n = 1..4096</a>
%H A323173 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A323173 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A323173 a(n) = A000203(A122111(n)).
%F A323173 a(n) = 2*A122111(n) - A323174(n).
%F A323173 a(n) = A322819(n) * A038712(A122111(n)).
%t A323173 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 A323173 a[n_] := With[{k = A122111[n]}, DivisorSigma[1, k]];
%t A323173 Array[a, 70] (* _Jean-François Alcover_, Sep 23 2020 *)
%o A323173 (PARI)
%o A323173 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 A323173 A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
%o A323173 A323173(n) = sigma(A122111(n));
%Y A323173 Cf. A000203, A038712, A122111, A322819, A323174.
%Y A323173 Cf. also A323243.
%K A323173 nonn
%O A323173 1,2
%A A323173 _Antti Karttunen_, Jan 10 2019