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.

A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 12, 4, 6, 16, 20, 12, 24, 24, 32, 5, 32, 12, 36, 24, 48, 40, 44, 16, 12, 48, 8, 36, 56, 64, 60, 6, 80, 64, 96, 18, 72, 72, 96, 32, 80, 96, 84, 60, 48, 88, 92, 20, 18, 24, 128, 72, 104, 16, 160, 48, 144, 112, 116, 96, 120, 120, 72, 7, 192, 160, 132, 96, 176, 192
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(20) = a(2^2*5) = (2 - 1)*(2 + 1) * (5 - 1)*(1 + 1) = 24.

Links

Crossrefs

Cf. A000005, A000010, A000026, A000040, A000302 (numbers n such that a(n) is odd), A001221, A006093, A007947, A023900, A034444, A059975, A062355, A173557, A304407, A304409, A304411, A304412.

Programs

  • Maple
    a:= n-> mul((i[1]-1)*(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021
  • Mathematica
    a[n_] := Times @@ ((#[[1]] - 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 70}]
Table[DivisorSigma[0, n] EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]], {n, 70}]
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*(e+1))} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*abs(A023900(n)) = A000005(n)*A173557(n) = A000005(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*phi(n) if n is a squarefree (A005117), where omega() = A001221 and phi() = A000010.

A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

Original entry on oeis.org

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.

Links

Crossrefs

Cf. A000005, A000026, A000040, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

Programs

  • Mathematica
    a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]
  • PARI
    a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*A007947(n).
a(p^k) = p*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 2/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, Jun 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(p^s - 1))^2.
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...
and gamma is the Euler-Mascheroni constant A001620. (End)

A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 9, 8, 18, 12, 24, 14, 24, 24, 12, 18, 24, 20, 36, 32, 36, 24, 36, 12, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.

Links

Crossrefs

Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.
Cf. A072691, A330596.

Programs

  • Mathematica
    a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
a(p^k) = (p + 1)*k where p is a prime and k > 0.
a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - Amiram Eldar, Nov 30 2022

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

Original entry on oeis.org

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.

Links

Crossrefs

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.
Cf. A065463.

Programs

  • Mathematica
    a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]
  • PARI
    a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...
and gamma is the Euler-Mascheroni constant A001620. (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.