A324546 - OEIS

%I #12 Mar 08 2019 20:14:57

%S 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 0,28,-12,30,1,12,14,22,-19,36,16,18,-10,40,-12,42,4,41,20,46,-28,41,

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

%C 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 Antti Karttunen, <a href="/A324546/b324546.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A324546/a324546.txt">Data supplement: n, a(n) computed for n = 1..65539</a>

%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A033879(A250246(n)) = 2*A250246(n) - A324545(n).

%F a(n) = A250246(n) - A324535(n).

%o (PARI)

%o up_to = 65539;

%o 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 A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639

%o A055396(n) = if(1==n,0,primepi(A020639(n)));

%o v078898 = ordinal_transform(vector(up_to,n,A020639(n)));

%o A078898(n) = v078898[n];

%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961

%o 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(n) = { my(k=A250246(n)); (k+k - sigma(k)); };

%Y Cf. A000396, A033879, A083221, A250246, A324535, A324545.

%Y Cf. also A323244, A323174, A324574, A324575.

%K sign

%O 1,3

%A _Antti Karttunen_, Mar 06 2019