A103977 Zumkeller deficiency of n: Let d_1 ... d_k be the divisors of n. Then a(n) = min_{ e_1 = +-1, ... e_k = +-1 } | Sum_i e_i d_i |.

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

Offset

1,3

Comments

Like the ordinary deficiency (A033879) obtains 0's only at perfect numbers (A000396), the Zumkeller deficiency obtains 0's only at integer-perfect numbers, A083207. See the formula section. Unlike the ordinary deficiency, this obtains only nonnegative values. See A378600 for another version. - Antti Karttunen, Dec 03 2024

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from Amiram Eldar)

Formula

If n=p (prime), then a(n)=p-1. If n=2^m, then a(n)=1. [Corrected by R. J. Mathar, Nov 27 2007]

a(n) = 0 iff n is a Zumkeller number (A083207). - Amiram Eldar, Jan 05 2020

From Antti Karttunen, Dec 03 2024: (Start)

a(n) = A033879(n) iff n is a non-abundant number (A263837).

a(n) = abs(A378600(n)).

a(n) = 2*A378647(n) - A378648(n). [Analogously to A033879(n) = 2*n - sigma(n)]

a(n) = 0 <=> A083206(n) > 0.

(End)

a(p^e) = p^e - (1+p+...+p^(e-1)) = (p^e*(p-2) + 1)/(p-1) for prime p. - Jianing Song, Dec 05 2024

a(n) = 1 <=> A379504(n) > 0. - Antti Karttunen, Jan 07 2025

Example

a(6) = 1 + 2 + 3 - 6 = 0.

Maple

A103977 := proc(n) local divs, a, acandid, filt, i, p, sigs ; divs := convert(numtheory[divisors](n), list) ; a := add(i, i=divs) ; for sigs from 0 to 2^nops(divs)-1 do filt := convert(sigs, base, 2) ; while nops(filt) < nops(divs) do filt := [op(filt), 0] ; od ; acandid := 0 ; for p from 0 to nops(divs)-1 do if op(p+1, filt) = 0 then acandid := acandid-op(p+1, divs) ; else acandid := acandid+op(p+1, divs) ; fi ; od: acandid := abs(acandid) ; if acandid < a then a := acandid ; fi ; od: RETURN(a) ; end: seq(A103977(n), n=1..80) ; # R. J. Mathar, Nov 27 2007
# second Maple program:
a:= proc(n) option remember; local l, b; l, b:= [numtheory[divisors](n)[]],
      proc(s, i) option remember; `if`(i<1, s,
        min(b(s+l[i], i-1), b(abs(s-l[i]), i-1)))
      end: b(0, nops(l))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Dec 05 2024

Mathematica

a[n_] := Module[{d = Divisors[n], c, p, m}, c = CoefficientList[Product[1 + x^i, {i, d}], x]; p = -1 + Position[c, _?(# > 0 &)] // Flatten; m = Length[p]; If[OddQ[m], If[(d = p[[(m + 1)/2]] - p[[(m - 1)/2]]) == 1, 0, d], p[[m/2 + 1]] - p[[m/2]]]]; Array[a, 100] (* Amiram Eldar, Dec 11 2019 *)

Prog

(PARI)
nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1, #v, if(v[i], listput(lista, i))); Vec(lista); }; \\ Doesn't need to be 0-based, as we use their differences only.
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)); }; \\ Antti Karttunen, Dec 03 2024, after Mathematica-program by Amiram Eldar

Crossrefs

Cf. A125732, A125733, A005835, A023196, A033879, A083206, A083207 (positions of 0's), A263837, A378643 (Dirichlet inverse), A378644 (Möbius transform), A378645, A378646, A378647 (an analog of A000027), A378648 (an analog of sigma), A378649 (an analog of Euler phi), A379503 (positions of 1's), A379504, A379505.

Cf. A378600 (signed variant).

Cf. also A058377, A119347.

Sequence in context: A326144 A120112 A233150 * A378600 A109883 A033880

Adjacent sequences: A103974 A103975 A103976 * A103978 A103979 A103980

Keyword

nonn

Author

Yasutoshi Kohmoto, Jan 01 2007

Extensions

More terms from R. J. Mathar, Nov 27 2007

Name "Zumkeller deficiency" coined by Antti Karttunen, Dec 03 2024

Status

approved