cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A378653 a(n) = A378648(n) - sigma(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 20, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 36, 0, 0, 0, 14, 0, 12, 0, 0, 0, 0, 0, 68, 0, 0, 0, 0, 0, 20, 0, 8, 0, 0, 0, 84, 0, 0, 0, 0, 0, 12, 0, 0, 0, 4, 0, 156, 0, 0, 0, 0, 0, 12, 0, 54, 0, 0, 0, 88, 0, 0, 0, 4, 0, 86, 0, 0, 0, 0, 0, 196, 0, 0, 0, 22, 0, 12, 0, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Dec 03 2024

Keywords

Links

Crossrefs

Inverse Möbius transform of A378650.
Cf. A000203, A378648.
Cf. A005101 (positions of nonzero terms), A263837 (of 0's).

Programs

Formula

a(n) = A378648(n) - A000203(n).
a(n) = Sum_{d|n} A378650(d).

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 |.

Original entry on oeis.org

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

Views

Author

Yasutoshi Kohmoto, Jan 01 2007

Keywords

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

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

A378647 Dirichlet convolution of A074206 and A103977, where A074206 is the number of ordered factorizations of n, and A103977 is the Zumkeller deficiency of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13, 14, 15, 16, 17, 22, 19, 22, 21, 22, 23, 40, 25, 26, 27, 28, 29, 42, 31, 32, 33, 34, 35, 64, 37, 38, 39, 52, 41, 54, 43, 44, 45, 46, 47, 96, 49, 50, 51, 52, 53, 70, 55, 64, 57, 58, 59, 126, 61, 62, 63, 64, 65, 78, 67, 68, 69, 74, 71, 176, 73, 74, 75, 76, 77, 90, 79, 120, 81
Offset: 1

Views

Author

Antti Karttunen, Dec 03 2024

Keywords

Comments

Möbius transform of A378648, which is the Dirichlet convolution of A067824 and A103977.

Links

Crossrefs

Cf. A000027, A008683, A067824, A074206, A103977, A263837, A378648 (inverse Möbius transform), A378649 (Möbius transform), A378650 [= a(n)-n], A378655 (Dirichlet inverse).

Programs

Formula

a(n) = Sum_{d|n} A074206(d)*A103977(n/d).
a(n) = Sum_{d|n} A008683(d)*A378648(n/d).
a(n) = Sum_{d|n} A067824(d)*A378644(n/d).
a(n) = A378650(n)+n, with a(n) = n if and only if n is a non-abundant number (A263837).
Showing 1-3 of 3 results.

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