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A324546 An analog of deficiency (A033879) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

%I A324546 #12 Mar 08 2019 20:14:57
%S A324546 1,1,2,1,4,0,6,1,5,2,10,-4,12,4,6,1,16,-3,18,-2,14,8,22,-12,19,10,10,
%T A324546 0,28,-12,30,1,12,14,22,-19,36,16,18,-10,40,-12,42,4,41,20,46,-28,41,
%U A324546 7,26,6,52,-12,94,-8,22,26,58,-48,60,28,22,1,38,-54,66,10,30,-4,70,-51,72,34,30,12,58,-12,78,-26,42,38,82,-64,102,40,18,-4,88
%N A324546 An analog of deficiency (A033879) for nonstandard factorization based on the sieve of Eratosthenes (A083221).
%C A324546 Even positions for zeros is given by the even terms of A000396, because they are among the fixed points of permutation A250246. Whether there are any zeros in odd positions depends on whether there are any odd perfect numbers. If such zeros exist, they would not necessarily be in the same positions as in A033879.
%H A324546 Antti Karttunen, <a href="/A324546/b324546.txt">Table of n, a(n) for n = 1..16384</a>
%H A324546 Antti Karttunen, <a href="/A324546/a324546.txt">Data supplement: n, a(n) computed for n = 1..65539</a>
%H A324546 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%H A324546 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A324546 a(n) = A033879(A250246(n)) = 2*A250246(n) - A324545(n).
%F A324546 a(n) = A250246(n) - A324535(n).
%o A324546 (PARI)
%o A324546 up_to = 65539;
%o A324546 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A324546 A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
%o A324546 A055396(n) = if(1==n,0,primepi(A020639(n)));
%o A324546 v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
%o A324546 A078898(n) = v078898[n];
%o A324546 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A324546 A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
%o A324546 A324546(n) = { my(k=A250246(n)); (k+k - sigma(k)); };
%Y A324546 Cf. A000396, A033879, A083221, A250246, A324535, A324545.
%Y A324546 Cf. also A323244, A323174, A324574, A324575.
%K A324546 sign
%O A324546 1,3
%A A324546 _Antti Karttunen_, Mar 06 2019