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

Author

Yasutoshi Kohmoto, Jan 01 2007

Keywords

• nonn

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

Examples

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

Links

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

Programs

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 *)

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

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

Extensions

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

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