A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 0, 12, 4, 6, 1, 16, -1, 18, 0, 10, 8, 22, 0, 19, 10, 14, 0, 28, 0, 30, 1, 18, 14, 22, -1, 36, 16, 22, 0, 40, 0, 42, 4, 12, 20, 46, 0, 41, 7, 30, 6, 52, 0, 38, 0, 34, 26, 58, 0, 60, 28, 22, 1, 46, 0, 66, 10, 42, 0, 70, -1, 72, 34, 26, 12, 58, 0, 78, 0, 41, 38, 82, 0, 62, 40, 54, 0, 88, 0, 70

Offset: 1

Author

Antti Karttunen, Dec 04 2024

Keywords

• sign

Comments

If n is abundant, then negate the value of A103977(n), otherwise use as it is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A033879, A103977.

Cf. A005100 (positions of terms > 0), A083207 (positions of 0's), A083211 (positions of negative terms), A156903 (positions of odd negative terms), A171641 (of even negative terms).

Programs

PARI

A033879(n) = (n+n-sigma(n));
nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); };
A103977(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); my(plist=nonzerocoefpositions(p), m = #plist, d); if(!(m%2), plist[1+(m/2)]-plist[m/2], d = plist[(m+1)/2]-plist[(m-1)/2]; if(1==d,0,d)); };
A378600(n) = { my(d=A033879(n)); if(d>=0, d, -A103977(n)); };

Formula

If A033879(n) >= 0, a(n) = A033879(n), otherwise a(n) = -A103977(n).