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 1-5 of 5 results.

A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 12, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 36, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 03 2020

Keywords

Examples

			a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.

Links

Crossrefs

Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.

Programs

  • Mathematica
    a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA001221(n/d) * A003557(n/d) * a(d).
a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).
a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - Antti Karttunen, Oct 29 2021
Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - Amiram Eldar, Dec 07 2023

A347960 Numbers k for which A348036(k) > A007947(k).

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 900, 936, 968, 972, 980, 1000, 1008, 1044, 1080, 1089, 1100, 1116, 1125, 1152, 1156, 1176, 1188, 1200, 1224, 1260, 1296
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Comments

Numbers k such that A348039(k) > 1.
Numbers k such that A348037(k) < A003557(k).
Numbers k such that A327564(k) > A348038(k).
Differs from A036785 and A338539 for the first time at n=20, where a(n) = 450, as A036785(20) = A338539(20) = 441 is not included in this sequence.

Crossrefs

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038, A348039.

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); q[n_] := GCD[n, Times @@ f @@@ (fct = FactorInteger[n])] > Times @@ First /@ fct; Select[Range[1300], q] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
isA347960(n) = (A348036(n)>A007947(n));

A348036 a(n) = gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 72, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Links

Crossrefs

Differs from A007947 at the positions given by A347960.
Cf. A003959, A003968, A348030, A348037, A348038, A348039.

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348036(n) = gcd(n, A003968(n));

Formula

a(n) = gcd(n, A003968(n)).
a(n) = gcd(n, A348030(n)) = gcd(A003968(n), A348030(n)).
a(n) = n / A348037(n) = A003968(n) / A348038(n).
a(n) = A007947(n) * A348039(n).

A348037 a(n) = n / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 5, 1, 1, 1, 4, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Links

Crossrefs

Differs from A003557 at the positions given by A347960.
Cf. A003959, A003968, A333634, A348038, A348039, A348499 (positions of 1's).

Programs

  • Mathematica
    f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := n / GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)
  • PARI
    A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348037(n) = (n/gcd(n, A003968(n)));

Formula

a(n) = n / A348036(n) = n / gcd(n, A003968(n)).
a(n) = A003557(n) / A348039(n).

A348039 a(n) = gcd(A003557(n), A327564(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6
Offset: 1

Views

Author

Antti Karttunen, Oct 19 2021

Keywords

Comments

There is interesting regularity in the scatter plot.

Links

Crossrefs

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038.
Cf. A347960 (positions of terms > 1).

Programs

  • Mathematica
    {1}~Join~Array[GCD @@ Map[Times @@ # &, Transpose@ Map[{#1^(#2 - 1), (#1 + 1)^(#2 - 1)} & @@ # &, FactorInteger[#]]] &, 105, 2] (* Michael De Vlieger, Oct 20 2021 *)
  • PARI
    A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
A348039(n) = (A348036(n)/A007947(n));
  • PARI
    A003557(n) = (n/factorback(factorint(n)[, 1]));
A327564(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1]++; f[k, 2]--); factorback(f); }; \\ From A327564
A348039(n) = gcd(A003557(n), A327564(n));

Formula

a(n) = gcd(A003557(n), A327564(n)).
a(n) = A348036(n) / A007947(n).
a(n) = A003557(n) / A348037(n).
a(n) = A327564(n) / A348038(n).
Showing 1-5 of 5 results.

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