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.

A347090 Dirichlet inverse of A332844.

Original entry on oeis.org

1, -3, -4, 1, -6, 12, -8, 3, 2, 18, -12, -4, -14, 24, 24, -2, -18, -6, -20, -6, 32, 36, -24, -12, 4, 42, 4, -8, -30, -72, -32, 0, 48, 54, 48, 2, -38, 60, 56, -18, -42, -96, -44, -12, -12, 72, -48, 8, 6, -12, 72, -14, -54, -12, 72, -24, 80, 90, -60, 24, -62, 96, -16, 0, 84, -144, -68, -18, 96, -144, -72, 6, -74, 114
Offset: 1

Views

Author

Antti Karttunen, Aug 18 2021

Keywords

Comments

Multiplicative because A332844 is.

Links

Crossrefs

Cf. A332844, A347091.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
v347090 = DirInverseCorrect(vector(up_to,n,A332844(n)));
A347090(n) = v347090[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA332844(n/d) * a(d).
a(n) = A347091(n) - A332844(n).

A344442 a(n) = A332844(n) - n.

Original entry on oeis.org

0, 1, 1, 4, 1, 6, 1, 10, 5, 8, 1, 20, 1, 10, 9, 23, 1, 24, 1, 28, 11, 14, 1, 48, 7, 16, 17, 36, 1, 42, 1, 49, 15, 20, 13, 76, 1, 22, 17, 68, 1, 54, 1, 52, 39, 26, 1, 108, 9, 46, 21, 60, 1, 78, 17, 88, 23, 32, 1, 132, 1, 34, 49, 102, 19, 78, 1, 76, 27, 74, 1, 180, 1, 40, 53, 84, 19, 90, 1, 154, 54, 44, 1, 172, 23, 46
Offset: 1

Views

Author

Antti Karttunen, May 23 2021

Keywords

Links

Crossrefs

Cf. A000010, A124315, A332844.

Programs

  • Mathematica
    Table[DivisorSum[n, Boole[IntegerQ[(n/#)^(1/2)]] DivisorSigma[1, #] &] - n, {n, 86}] (* Michael De Vlieger, May 24 2021 *)
  • PARI
    A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
A344442(n) = (A332844(n) - n);

Formula

a(n) = A332844(n) - n.
a(n) = Sum_{d|n} A000010(n/d) * (A124315(d)-1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^6/540 - 1 = 0.7803503... . - Amiram Eldar, Dec 04 2024

A347091 Sum of A332844 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 21, 16, 36, 0, 28, 0, 48, 48, 37, 0, 36, 0, 42, 64, 72, 0, 60, 36, 84, 48, 56, 0, 0, 0, 81, 96, 108, 96, 114, 0, 120, 112, 90, 0, 0, 0, 84, 72, 144, 0, 164, 64, 84, 144, 98, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 96, 166, 168, 0, 0, 126, 192, 0, 0, 258, 0, 228, 112, 140, 192, 0, 0, 246, 132
Offset: 1

Views

Author

Antti Karttunen, Aug 18 2021

Keywords

Comments

No negative terms in range 1 .. 2^20.
Apparently, A030059 gives the positions of all zeros.

Links

Crossrefs

Cf. A030059, A030229, A332844, A347090.
Cf. also A347093.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
v347090 = DirInverseCorrect(vector(up_to,n,A332844(n)));
A347090(n) = v347090[n];
A347091(n) = (A332844(n)+A347090(n));

Formula

a(n) = A332844(n) + A347090(n).
For n > 1, a(n) = -Sum_{d|n, 1A332844(d) * A347090(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A332844(A030229(n)).
Showing 1-3 of 3 results.

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