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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 |.

%I A103977 #45 Jan 07 2025 19:05:52
%S A103977 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,
%T A103977 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,
%U A103977 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
%N 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 |.
%C A103977 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
%H A103977 Antti Karttunen, <a href="/A103977/b103977.txt">Table of n, a(n) for n = 1..20000</a> (first 10000 terms from Amiram Eldar)
%F A103977 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]
%F A103977 a(n) = 0 iff n is a Zumkeller number (A083207). - _Amiram Eldar_, Jan 05 2020
%F A103977 From _Antti Karttunen_, Dec 03 2024: (Start)
%F A103977 a(n) = A033879(n) iff n is a non-abundant number (A263837).
%F A103977 a(n) = abs(A378600(n)).
%F A103977 a(n) = 2*A378647(n) - A378648(n). [Analogously to A033879(n) = 2*n - sigma(n)]
%F A103977 a(n) = 0 <=> A083206(n) > 0.
%F A103977 (End)
%F A103977 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
%F A103977 a(n) = 1 <=> A379504(n) > 0. - _Antti Karttunen_, Jan 07 2025
%e A103977 a(6) = 1 + 2 + 3 - 6 = 0.
%p A103977 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
%p A103977 # second Maple program:
%p A103977 a:= proc(n) option remember; local l, b; l, b:= [numtheory[divisors](n)[]],
%p A103977       proc(s, i) option remember; `if`(i<1, s,
%p A103977         min(b(s+l[i], i-1), b(abs(s-l[i]), i-1)))
%p A103977       end: b(0, nops(l))
%p A103977     end:
%p A103977 seq(a(n), n=1..80);  # _Alois P. Heinz_, Dec 05 2024
%t A103977 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 *)
%o A103977 (PARI)
%o A103977 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.
%o A103977 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_
%Y A103977 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.
%Y A103977 Cf. A378600 (signed variant).
%Y A103977 Cf. also A058377, A119347.
%K A103977 nonn
%O A103977 1,3
%A A103977 _Yasutoshi Kohmoto_, Jan 01 2007
%E A103977 More terms from _R. J. Mathar_, Nov 27 2007
%E A103977 Name "Zumkeller deficiency" coined by _Antti Karttunen_, Dec 03 2024