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

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

Offset: 1

Author

Antti Karttunen, Mar 06 2019

Keywords

• sign

Comments

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539
• Index entries for sequences generated by sieves
• Index entries for sequences related to sigma(n)

Crossrefs

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

Cf. also A323244, A323174, A324574, A324575.

Programs

PARI

up_to = 65539;
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)));
A324546(n) = { my(k=A250246(n)); (k+k - sigma(k)); };

Formula

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

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