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 1-4 of 4 results.

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2019

Keywords

Links

Crossrefs

Cf. A033879, A323911, A323912, A359549 (parity of terms).
Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

Programs

  • Mathematica
    b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)
  • PARI
    up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

Original entry on oeis.org

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55
Offset: 1

Views

Author

Alonso del Arte, May 26 2011

Keywords

Comments

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Links

Crossrefs

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).
Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

Programs

  • Mathematica
    hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)
  • PARI
    a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

Formula

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011
From Antti Karttunen, Nov 22 2024: (Start)
Following formulas were conjectured by Sequence Machine:
For n > 1, a(n) = A191150(n) + A074206(n).
a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.
a(n) = Sum_{d|n} A330575(d).
a(n) = Sum_{d|n} d*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A074206(n/d).
a(n) = Sum_{d|n} A007429(d)*A174725(n/d).
a(n) = Sum_{d|n} A000010(d)*A253249(n/d).
a(n) = Sum_{d|n} A038040(d)*A323912(n/d).
a(n) = Sum_{d|n} A060640(d)*A323910(n/d).
(End)

A323913 Sum of A083254 (2*phi(n) - n) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, -4, 0, 0, 10, 0, 0, 0, 9, 0, 5, 0, 0, -16, 0, 0, 18, 0, 30, -4, 0, 0, 22, 0, 0, -24, 0, 0, 11, 0, 0, 0, 25, -12, 30, 0, 0, -18, 54, 0, 34, 0, 0, -24, 0, 0, 21, 0, 66, -40, 0, 0, 42, -32, 0, 0, 0, 0, 9, 0, 90, -48, 0, 0, 19, 0, 0, -40, 90, 0, 54, 0, 0, -38, 110, 0, 58, 0, 102, 0, 0, -20
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2019

Keywords

Links

Crossrefs

Cf. A000010, A083254, A319340, A323911, A323912.

Programs

  • PARI
    up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];
A323913(n) = (A083254(n)+A323912(n));

A376404 Dirichlet inverse of 2*phi(n) - phi(A003961(n)), where phi is Euler totient function and A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 0, 0, 2, -2, 4, -2, 10, 8, 4, -8, 16, -8, 8, 8, 42, -14, 28, -14, 12, 16, 4, -16, 72, 6, 8, 64, 28, -26, 16, -24, 170, 8, 4, 20, 144, -32, 8, 16, 52, -38, 40, -38, 0, 40, 12, -40, 328, 30, 28, 8, 16, -46, 228, 24, 124, 16, 4, -56, 112, -54, 12, 96, 682, 32, -8, -62, -12, 24, 24, -68, 712, -66, 8, 56, 4, 32, 16
Offset: 1

Views

Author

Antti Karttunen, Nov 15 2024

Keywords

Links

Crossrefs

Cf. A000010, A003961, A003972.
Dirichlet inverse of -A349754, inverse Möbius transform of A346248.
Cf. also A323912.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349754(n) = (eulerphi(A003961(n))-2*eulerphi(n));
memoA376404 = Map();
A376404(n) = if(1==n,1,my(v); if(mapisdefined(memoA376404,n,&v), v, v = -sumdiv(n,d,if(dA349754(n/d)*A376404(d),0)); mapput(memoA376404,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA349754(n/d) * a(d).
a(n) = Sum_{d|n} A346248(d).
Showing 1-4 of 4 results.

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