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 1-8 of 8 results.

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

A166629 Totally multiplicative sequence with a(p) = 8p for prime p.

Original entry on oeis.org

1, 16, 24, 256, 40, 384, 56, 4096, 576, 640, 88, 6144, 104, 896, 960, 65536, 136, 9216, 152, 10240, 1344, 1408, 184, 98304, 1600, 1664, 13824, 14336, 232, 15360, 248, 1048576, 2112, 2176, 2240, 147456, 296, 2432, 2496, 163840, 328, 21504, 344, 22528
Offset: 1

Views

Author

Jaroslav Krizek, Oct 18 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829.

Programs

  • Mathematica
    Table[n*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, May 19 2016 *)
  • PARI
    a(n) = n*8^bigomega(n); \\ Michel Marcus, Oct 30 2019

Formula

Multiplicative with a(p^e) = (8p)^e.
If n = Product p(k)^e(k) then a(n) = Product (8*p(k))^e(k).
a(n) = n * A165829(n) = n * 8^bigomega(n) = n * 8^A001222(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 8 * p^(1 - s)). - Ilya Gutkovskiy, Oct 30 2019

A166638 Totally multiplicative sequence with a(p) = 8*(p-1) for prime p.

Original entry on oeis.org

1, 8, 16, 64, 32, 128, 48, 512, 256, 256, 80, 1024, 96, 384, 512, 4096, 128, 2048, 144, 2048, 768, 640, 176, 8192, 1024, 768, 4096, 3072, 224, 4096, 240, 32768, 1280, 1024, 1536, 16384, 288, 1152, 1536, 16384, 320, 6144, 336, 5120, 8192, 1408, 368, 65536
Offset: 1

Views

Author

Jaroslav Krizek, Oct 18 2009

Keywords

Links

Crossrefs

Cf. A003958, A001222, A165829.

Programs

  • Mathematica
    DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] :=
DirichletInverse[f][n] = -1/f[1]*Sum[f[n/d]*DirichletInverse[f][d], {d, Most[Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; a[m_] := DirichletInverse[muphi][m]; Table[a[m]*8^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (8*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

Formula

Multiplicative with a(p^e) = (8*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)-1))^e(k).
a(n) = A165829(n) * A003958(n) = 8^bigomega(n) * A003958(n) = 8^A001222(n) * A003958(n).

A166648 Totally multiplicative sequence with a(p) = 8*(p+1) for prime p.

Original entry on oeis.org

1, 24, 32, 576, 48, 768, 64, 13824, 1024, 1152, 96, 18432, 112, 1536, 1536, 331776, 144, 24576, 160, 27648, 2048, 2304, 192, 442368, 2304, 2688, 32768, 36864, 240, 36864, 256, 7962624, 3072, 3456, 3072, 589824, 304, 3840, 3584, 663552, 336, 49152
Offset: 1

Views

Author

Jaroslav Krizek, Oct 18 2009

Keywords

Links

Crossrefs

Cf. A001222, A003959, A165829.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*8^(PrimeOmega[n]), {n, 1, 100}] (* G. C. Greubel, May 21 2016 *)
f[p_, e_] := (8*(p+1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)
  • PARI
    a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = 8*(f[k,1]+1)); factorback(f);} \\ Michel Marcus, May 21 2016

Formula

Multiplicative with a(p^e) = (8*(p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)+1))^e(k).
a(n) = A165829(n) * A003959(n) = 8^bigomega(n) * A003959(n) = 8^A001222(n) * A003959(n).

A167300 Totally multiplicative sequence with a(p) = 8*(p-2) for prime p.

Original entry on oeis.org

1, 0, 8, 0, 24, 0, 40, 0, 64, 0, 72, 0, 88, 0, 192, 0, 120, 0, 136, 0, 320, 0, 168, 0, 576, 0, 512, 0, 216, 0, 232, 0, 576, 0, 960, 0, 280, 0, 704, 0, 312, 0, 328, 0, 1536, 0, 360, 0, 1600, 0, 960, 0, 408, 0, 1728, 0, 1088, 0, 456, 0, 472, 0, 2560, 0, 2112, 0
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166586.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (8*(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) = (8*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)-2))^e(k).
a(2k) = 0 for k >= 1.
a(n) = A165829(n) * A166586(n) = 8^bigomega(n) * A166586(n) = 8^A001222(n) * A166586(n).

A167309 Totally multiplicative sequence with a(p) = 8*(p+2) for prime p.

Original entry on oeis.org

1, 32, 40, 1024, 56, 1280, 72, 32768, 1600, 1792, 104, 40960, 120, 2304, 2240, 1048576, 152, 51200, 168, 57344, 2880, 3328, 200, 1310720, 3136, 3840, 64000, 73728, 248, 71680, 264, 33554432, 4160, 4864, 4032, 1638400, 312, 5376, 4800, 1835008, 344, 92160, 360
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166590.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 07 2016 *)
f[p_, e_] := (8*(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) = (8*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)+2))^e(k).
a(n) = A165829(n) * A166590(n) = 8^bigomega(n) * A166590(n) = 8^A001222(n) * A166590(n).

A167318 Totally multiplicative sequence with a(p) = 8*(p-3) for prime p.

Original entry on oeis.org

1, -8, 0, 64, 16, 0, 32, -512, 0, -128, 64, 0, 80, -256, 0, 4096, 112, 0, 128, 1024, 0, -512, 160, 0, 256, -640, 0, 2048, 208, 0, 224, -32768, 0, -896, 512, 0, 272, -1024, 0, -8192, 304, 0, 320, 4096, 0, -1280, 352, 0, 1024, -2048, 0, 5120, 400, 0, 1024, -16384
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166589.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (8*(p-3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2023 *)

Formula

Multiplicative with a(p^e) = (8*(p-3))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)-3))^e(k).
a(3k) = 0 for k >= 1.
a(n) = A165829(n) * A166589(n) = 8^bigomega(n) * A166589(n) = 8^A001222(n) * A166589(n).

A167327 Totally multiplicative sequence with a(p) = 8*(p+3) for prime p.

Original entry on oeis.org

1, 40, 48, 1600, 64, 1920, 80, 64000, 2304, 2560, 112, 76800, 128, 3200, 3072, 2560000, 160, 92160, 176, 102400, 3840, 4480, 208, 3072000, 4096, 5120, 110592, 128000, 256, 122880, 272, 102400000, 5376, 6400, 5120, 3686400, 320, 7040, 6144, 4096000, 352, 153600
Offset: 1

Views

Author

Jaroslav Krizek, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A001222, A165829, A166591.

Programs

  • Mathematica
    a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*8^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, Jun 09 2016 *)
f[p_, e_] := (8*(p+3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 21 2023 *)

Formula

Multiplicative with a(p^e) = (8*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product (8*(p(k)+3))^e(k).
a(n) = A165829(n) * A166591(n) = 8^bigomega(n) * A166591(n) = 8^A001222(n) * A166591(n).
Showing 1-8 of 8 results.

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

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