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 1-10 of 20 results. Next

A003959 If n = Product p(k)^e(k) then a(n) = Product (p(k)+1)^e(k), a(1) = 1.

Original entry on oeis.org

1, 3, 4, 9, 6, 12, 8, 27, 16, 18, 12, 36, 14, 24, 24, 81, 18, 48, 20, 54, 32, 36, 24, 108, 36, 42, 64, 72, 30, 72, 32, 243, 48, 54, 48, 144, 38, 60, 56, 162, 42, 96, 44, 108, 96, 72, 48, 324, 64, 108, 72, 126, 54, 192, 72, 216, 80, 90, 60, 216, 62, 96, 128, 729, 84, 144, 68
Offset: 1

Views

Author

Marc LeBrun

Keywords

Comments

Completely multiplicative.
Sum of divisors of n with multiplicity. If n = p^m, the number of ways to make p^k as a divisor of n is C(m,k); and sum(C(m,k)*p^k) = (p+1)^k. The rest follows because the function is multiplicative. - Franklin T. Adams-Watters, Jan 25 2010

Links

Crossrefs

Apart from initial terms, same as A064478.
Cf. A003958, A027746, A036987, A061142, A163407.
Cf. A063441, A165824, A165825, A165826, A165827.
Cf. A165828, A165829, A165831, A167350, A166590.
Cf. A166642, A166643, A166644, A166645, A166646.
Cf. A166647, A166649, A166650, A166586, A165830.
Cf. A167351, A168065, A168066.

Programs

  • Haskell
    a003959 1 = 1
a003959 n = product $ map (+ 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
  • Maple
    a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017
  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]]+1)^fi[[All, 2]])); a /@ Range[67] (* Jean-François Alcover, Apr 22 2011 *)
  • PARI
    a(n)=if(n<1,0,direuler(p=2,n,1/(1-X-p*X))[n]) /* Ralf Stephan */

Formula

Multiplicative with a(p^e) = (p+1)^e. - David W. Wilson, Aug 01 2001
Sum_{n>0} a(n)/n^s = Product_{p prime} 1/(1-p^(-s)-p^(1-s)) (conjectured). - Ralf Stephan, Jul 07 2013
This follows from the absolute convergence of the sum (compare with a(n) = n^2) and the Euler product for completely multiplicative functions. Convergence occurs for at least Re(s)>3. - Thomas Anton, Jul 15 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065488/2 = 1/(2*A005596) = 1.3370563627850107544802059152227440187511993141988459926... - Vaclav Kotesovec, Jul 17 2021
From Thomas Scheuerle, Jul 19 2021: (Start)
a(n) = gcd(A166642(n), A166643(n)).
a(n) = A166642(n)/A061142(n).
a(n) = A166643(n)/A165824(n).
a(n) = A166644(n)/A165825(n).
a(n) = A166645(n)/A165826(n).
a(n) = A166646(n)/A165827(n).
a(n) = A166647(n)/A165828(n).
a(n) = A166649(n)/A165830(n).
a(n) = A166650(n)/A165831(n).
a(n) = A167351(n)/A166590(n). (End)
Dirichlet g.f.: zeta(s-1) * Product_{primes p} (1 + 1/(p^s - p - 1)). - Vaclav Kotesovec, Aug 22 2021

Extensions

Definition reedited (with formula) by Daniel Forgues, Nov 17 2009

A166589 Totally multiplicative sequence with a(p) = p-3 for prime p.

Original entry on oeis.org

1, -1, 0, 1, 2, 0, 4, -1, 0, -2, 8, 0, 10, -4, 0, 1, 14, 0, 16, 2, 0, -8, 20, 0, 4, -10, 0, 4, 26, 0, 28, -1, 0, -14, 8, 0, 34, -16, 0, -2, 38, 0, 40, 8, 0, -20, 44, 0, 16, -4, 0, 10, 50, 0, 16, -4, 0, -26, 56, 0, 58, -28, 0, 1, 20, 0, 64, 14, 0, -8, 68, 0
Offset: 1

Views

Author

Jaroslav Krizek, Oct 17 2009

Keywords

Links

Crossrefs

Cf. A166586.

Programs

  • Mathematica
    a[1] = 1; a[p_?PrimeQ] := p-3; a[n_] := Times @@ Power @@@ ({#[[1]]-3, #[[2]]}& /@ FactorInteger[n]); Array[a, 72] (* Jean-François Alcover, Jul 19 2017 *)
  • PARI
    a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] -=3); factorback(f); \\ Michel Marcus, Jun 09 2014

Formula

Multiplicative with a(p^e) = (p-3)^e. If n = Product p(k)^e(k) then a(n) = Product (p(k)-3)^e(k). a(3k) = 0 for k >= 1. Abs (a(2^k)) = 1 for k >= 1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p*(p-1)/(p^2-p+3)) = 0.196347937547... . - Amiram Eldar, Jan 20 2024

Extensions

More terms from Michel Marcus, Jun 09 2014

A276833 Sum of mu(d)*phi(d) over divisors d of n.

Original entry on oeis.org

1, 0, -1, 0, -3, 0, -5, 0, -1, 0, -9, 0, -11, 0, 3, 0, -15, 0, -17, 0, 5, 0, -21, 0, -3, 0, -1, 0, -27, 0, -29, 0, 9, 0, 15, 0, -35, 0, 11, 0, -39, 0, -41, 0, 3, 0, -45, 0, -5, 0, 15, 0, -51, 0, 27, 0, 17, 0, -57, 0, -59, 0, 5, 0, 33, 0, -65, 0, 21, 0, -69, 0, -71, 0, 3, 0, 45, 0, -77, 0, -1, 0, -81, 0, 45, 0, 27, 0, -87, 0, 55, 0, 29, 0, 51, 0, -95, 0, 9
Offset: 1

Views

Author

Jurjen N.E. Bos, Sep 20 2016

Keywords

Comments

Discovered when incorrectly applying Mobius inversion formula.
a(n)*a(m) = a(n*m) if gcd(n,m)=1 (has a simple proof).
Strongly multiplicative: a(p^e) = 2 - p. - Charles R Greathouse IV, Oct 01 2019

Examples

			mu(d)*phi(d) = 1*1,-1*1,-1*2, 1*2 for d=1,2,3,6, so a(6) = 1*1-1*1-1*2+1*2 = 0.

Links

Crossrefs

For squarefree numbers, the absolute value is equal to A166586 (first exception at 25).
Cf. A097945.

Programs

  • Maple
    with(numtheory):seq(convert(map(x->2-x,factorset(n)),`*`),n=1..99); # Robert FERREOL, Mar 14 2020
  • Mathematica
    Table[Sum[MoebiusMu[d] EulerPhi[d], {d, Divisors[n]}], {n, 99}] (* Indranil Ghosh, Mar 10 2017 *)
a[1] = 1; a[n_] := Times @@ ((2 - First[#])& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)
  • PARI
    r=0;fordiv(n,d,r+=moebius(d)*eulerphi(d));r
  • PARI
    a(n) = sumdiv(n, d, moebius(d)*eulerphi(d)); \\ Michel Marcus, Sep 30 2016
  • PARI
    a(n)=my(f=factor(n)[,1]); prod(i=1,#f, 2-f[i]) \\ Charles R Greathouse IV, Oct 01 2019
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Formula

a(n) = Sum_{d|n} mu(d)*phi(d).
G.f.: Sum_{k>=1} mu(k)*phi(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Product_{p prime and p|n} (2-p). - Robert FERREOL, Mar 14 2020
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 - p^(1-s) + p^(-s)). - Vaclav Kotesovec, Jun 14 2020
a(n) = Sum_{k = 1..n} mu(lcm(k, n)/k). - Peter Bala, Jan 16 2024

A167350 Totally multiplicative sequence with a(p) = (p+1)*(p-2) = p^2-p-2 for prime p.

Original entry on oeis.org

1, 0, 4, 0, 18, 0, 40, 0, 16, 0, 108, 0, 154, 0, 72, 0, 270, 0, 340, 0, 160, 0, 504, 0, 324, 0, 64, 0, 810, 0, 928, 0, 432, 0, 720, 0, 1330, 0, 616, 0, 1638, 0, 1804, 0, 288, 0, 2160, 0, 1600, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A003959, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)

Formula

Multiplicative with a(p^e) = ((p+1)*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+1)*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A003959(n) * A166586(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 3/p^3 + 2/p^4) = 0.1140434597... . - Amiram Eldar, Dec 15 2022

A167294 Totally multiplicative sequence with a(p) = 2*(p-2) for prime p.

Original entry on oeis.org

1, 0, 2, 0, 6, 0, 10, 0, 4, 0, 18, 0, 22, 0, 12, 0, 30, 0, 34, 0, 20, 0, 42, 0, 36, 0, 8, 0, 54, 0, 58, 0, 36, 0, 60, 0, 70, 0, 44, 0, 78, 0, 82, 0, 24, 0, 90, 0, 100, 0, 60, 0, 102, 0, 108, 0, 68, 0, 114, 0, 118, 0, 40, 0, 132, 0, 130, 0, 84, 0, 138, 0, 142, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A061142, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*2^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (2*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Formula

Multiplicative with a(p^e) = (2*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (2*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A061142(n) * A166586(n) = 2^bigomega(n) * A166586(n) = 2^A001222(n) * A166586(n).

A167295 Totally multiplicative sequence with a(p) = 3*(p-2) for prime p.

Original entry on oeis.org

1, 0, 3, 0, 9, 0, 15, 0, 9, 0, 27, 0, 33, 0, 27, 0, 45, 0, 51, 0, 45, 0, 63, 0, 81, 0, 27, 0, 81, 0, 87, 0, 81, 0, 135, 0, 105, 0, 99, 0, 117, 0, 123, 0, 81, 0, 135, 0, 225, 0, 135, 0, 153, 0, 243, 0, 153, 0, 171, 0, 177, 0, 135, 0, 297, 0, 195, 0, 189, 0, 207
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165824, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*3^PrimeOmega[n], {n, 1, 100}](* G. C. Greubel, Jun 05 2016 *)
f[p_, e_] := (3*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

Formula

Multiplicative with a(p^e) = (3*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165824(n) * A166586(n) = 3^bigomega(n) * A166586(n) = 3^A001222(n) * A166586(n).

A167296 Totally multiplicative sequence with a(p) = 4*(p-2) for prime p.

Original entry on oeis.org

1, 0, 4, 0, 12, 0, 20, 0, 16, 0, 36, 0, 44, 0, 48, 0, 60, 0, 68, 0, 80, 0, 84, 0, 144, 0, 64, 0, 108, 0, 116, 0, 144, 0, 240, 0, 140, 0, 176, 0, 156, 0, 164, 0, 192, 0, 180, 0, 400, 0, 240, 0, 204, 0, 432, 0, 272, 0, 228, 0, 236, 0, 320, 0, 528, 0, 260, 0, 336
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165825, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*4^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)
f[p_, e_] := (4*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 18 2023 *)

Formula

Multiplicative with a(p^e) = (4*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165825(n) * A166586(n) = 4^bigomega(n) * A166586(n) = 4^A001222(n) * A166586(n).

A167297 Totally multiplicative sequence with a(p) = 5*(p-2) for prime p.

Original entry on oeis.org

1, 0, 5, 0, 15, 0, 25, 0, 25, 0, 45, 0, 55, 0, 75, 0, 75, 0, 85, 0, 125, 0, 105, 0, 225, 0, 125, 0, 135, 0, 145, 0, 225, 0, 375, 0, 175, 0, 275, 0, 195, 0, 205, 0, 375, 0, 225, 0, 625, 0, 375, 0, 255, 0, 675, 0, 425, 0, 285, 0, 295, 0, 625, 0, 825, 0, 325, 0, 525
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165826, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*5^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (5*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (5*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (5*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165826(n) * A166586(n) = 5^bigomega(n) * A166586(n) = 5^A001222(n) * A166586(n).

A167298 Totally multiplicative sequence with a(p) = 6*(p-2) for prime p.

Original entry on oeis.org

1, 0, 6, 0, 18, 0, 30, 0, 36, 0, 54, 0, 66, 0, 108, 0, 90, 0, 102, 0, 180, 0, 126, 0, 324, 0, 216, 0, 162, 0, 174, 0, 324, 0, 540, 0, 210, 0, 396, 0, 234, 0, 246, 0, 648, 0, 270, 0, 900, 0, 540, 0, 306, 0, 972, 0, 612, 0, 342, 0, 354, 0, 1080, 0, 1188, 0, 390
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165827, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*6^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (6*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)

Formula

Multiplicative with a(p^e) = (6*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (6*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165827(n) * A166586(n) = 6^bigomega(n) * A166586(n) = 6^A001222(n) * A166586(n).

A167299 Totally multiplicative sequence with a(p) = 7*(p-2) for prime p.

Original entry on oeis.org

1, 0, 7, 0, 21, 0, 35, 0, 49, 0, 63, 0, 77, 0, 147, 0, 105, 0, 119, 0, 245, 0, 147, 0, 441, 0, 343, 0, 189, 0, 203, 0, 441, 0, 735, 0, 245, 0, 539, 0, 273, 0, 287, 0, 1029, 0, 315, 0, 1225, 0, 735, 0, 357, 0, 1323, 0, 833, 0, 399, 0, 413, 0, 1715, 0, 1617, 0, 455
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165828, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*7^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (7*(p-2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (7*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (7*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165828(n) * A166586(n) = 7^bigomega(n) * A166586(n) = 7^A001222(n) * A166586(n).
Showing 1-10 of 20 results. Next

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

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