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 1-6 of 6 results.

A323363 Dirichlet inverse of Dedekind's psi, A001615.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Links

Crossrefs

Cf. A048250 (absolute values).
Cf. A001615, A008836, A323364, A323372.

Programs

  • Mathematica
    psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := a[n] = If[n == 1, 1, -Sum[psi[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 75] (* Jean-François Alcover, Feb 15 2020 *)
f[p_, e_] := (-1)^e * (p + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)
  • PARI
    A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} psi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019
From Amiram Eldar, Oct 14 2020: (Start)
Multiplicative with a(p^e) = (-1)^e * (p+1).
a(n) = A008836(n) * A048250(n). (End)
Dirichlet g.f.: zeta(2*s)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Dec 05 2022

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 44, 49, 50, 51, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 58, 62, 29, 65, 66, 67, 68, 69, 58, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 79
Offset: 1

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Comments

Restricted growth sequence transform of ordered pair [A003557(n), A323363(n)].
For all i, j:
a(i) = a(j) => A291751(i) = A291751(j),
a(i) = a(j) => A323364(i) = A323364(j).

Links

Crossrefs

Cf. A001615, A003557, A291750, A291751, A323363, A323364.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
v323372 = rgs_transform(vector(up_to, n, [A003557(n), A323363(n)]));
A323372(n) = v323372[n];

A323399 Sum of Jordan function J_2(n), A007434 and its Dirichlet inverse, A046970.

Original entry on oeis.org

2, 0, 0, 9, 0, 48, 0, 45, 64, 144, 0, 120, 0, 288, 384, 189, 0, 240, 0, 360, 768, 720, 0, 408, 576, 1008, 640, 720, 0, 0, 0, 765, 1920, 1728, 2304, 888, 0, 2160, 2688, 1224, 0, 0, 0, 1800, 1920, 3168, 0, 1560, 2304, 1872, 4608, 2520, 0, 1968, 5760, 2448, 5760, 5040, 0, 1728, 0, 5760, 3840, 3069, 8064, 0, 0, 4320, 8448, 0, 0, 3480, 0, 8208, 4992, 5400
Offset: 1

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Links

Crossrefs

Cf. A007434, A046970.
Cf. also A319340, A322581, A323364.

Programs

  • PARI
    A007434(n) = sumdiv(n, d, d*d*moebius(n/d));
A046970(n) = if(1==n,n,my(f=factor(n)); for(i=1, #f~, f[i,1] = 1-(f[i,1]^2)); factorback(f[,1]));
A323399(n) = (A007434(n) + A046970(n));

Formula

a(n) = A007434(n) + A046970(n).

A323401 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A323363(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 23, 25, 3, 26, 27, 28, 3, 29, 3, 30, 31, 32, 3, 33, 34, 35, 32, 36, 3, 37, 32, 38, 39, 40, 3, 41, 3, 42, 43, 44, 45, 46, 3, 47, 42, 46, 3, 48, 3, 49, 50, 51, 42, 52, 3, 53, 54, 55, 3, 56, 57, 58, 59, 60, 3, 61, 62, 63, 64, 65, 59, 66, 3, 67, 68, 69, 3, 70, 3
Offset: 1

Views

Author

Antti Karttunen, Jan 15 2019

Keywords

Comments

Restricted growth sequence transform of function f, defined as f(n) = A323372(n) for all other numbers n, except f(p) = 0 for odd primes p.
For all i, j:
A323400(i) = A323400(j) => a(i) = a(j),
a(i) = a(j) => A322588(i) = A322588(j),
a(i) = a(j) => A323364(i) = A323364(j).

Links

Crossrefs

Cf. A001615, A322588, A323363, A323364, A323400,
Cf. also A323371, A323372.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
Aux323401(n) = if((n>2)&&isprime(n), 0, [A003557(n), A323363(n)]);
v323401 = rgs_transform(vector(up_to, n, Aux323401(n)));
A323401(n) = v323401[n];

A323403 Sum of sigma and its Dirichlet inverse: a(n) = A000203(n) + A046692(n).

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 15, 16, 36, 0, 20, 0, 48, 48, 31, 0, 30, 0, 30, 64, 72, 0, 60, 36, 84, 40, 40, 0, 0, 0, 63, 96, 108, 96, 97, 0, 120, 112, 90, 0, 0, 0, 60, 60, 144, 0, 124, 64, 78, 144, 70, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 80, 127, 168, 0, 0, 90, 192, 0, 0, 195, 0, 228, 104, 100, 192, 0, 0, 186, 121, 252, 0, 288, 216, 264, 240, 180, 0, 288
Offset: 1

Views

Author

Antti Karttunen, Jan 15 2019

Keywords

Links

Crossrefs

Cf. A000203, A046692.
Cf. also A319340, A322581, A323364, A323399, A323408.

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(dA047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v046692 = DirInverse(vector(up_to,n,sigma(n)));
A046692(n) = v046692[n];
A323403(n) = (sigma(n)+A046692(n));

Formula

a(n) = A000203(n) + A046692(n).

A323408 Sum of unitary phi and its Dirichlet inverse: a(n) = A047994(n) + A323407(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 5, 4, 8, 0, 10, 0, 12, 16, 15, 0, 12, 0, 20, 24, 20, 0, 18, 16, 24, 24, 30, 0, 0, 0, 35, 40, 32, 48, 32, 0, 36, 48, 36, 0, 0, 0, 50, 48, 44, 0, 30, 36, 32, 64, 60, 0, 28, 80, 54, 72, 56, 0, 8, 0, 60, 72, 71, 96, 0, 0, 80, 88, 0, 0, 64, 0, 72, 64, 90, 120, 0, 0, 60, 88, 80, 0, 12, 128, 84, 112, 90, 0, 16, 144, 110, 120
Offset: 1

Views

Author

Antti Karttunen, Jan 15 2019

Keywords

Links

Crossrefs

Cf. A047994, A323407.
Cf. also A319340, A322581, A323364, A323399, A323403.

Programs

  • PARI
    up_to = 65537;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v323407 = DirInverse(vector(up_to,n,A047994(n)));
A323407(n) = v323407[n];
A323408(n) = (A047994(n) + A323407(n));
Showing 1-6 of 6 results.

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