A165825 -id:A165825 - cp's OEIS frontend

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

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

Marc LeBrun

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

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Index to divisibility sequences

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.

a003959 1 = 1
a003959 n = product $ map (+ 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012

a:= n-> mul((i[1]+1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

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 *)

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X-p*X))[n]) /* Ralf Stephan */

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

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

A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 8, 2, 2, 4, 2, 2, 4, 8, 2, 8, 2, 4, 4, 2, 2, 4, 8, 2, 16, 4, 2, 4, 2, 8, 4, 2, 4, 16, 2, 2, 4, 4, 2, 4, 2, 4, 16, 2, 2, 16, 8, 8, 4, 4, 2, 16, 4, 4, 4, 2, 2, 8, 2, 2, 16, 16, 4, 4, 2, 4, 4, 4, 2, 16, 2, 2, 16, 4, 4, 4, 2, 16, 128, 2, 2

Offset: 1

Gus Wiseman, Feb 03 2018

			Sequence begins: 1, 1, 3/2, 3/2, 5/2, 3/2, 7/2, 5/2, 27/8, 5/2, 11/2, 9/4, 13/2, 7/2.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Andrew Howroyd)

Cf. A000010, A003958, A005187, A006519, A007431, A011371, A018804, A023900, A029935, A037445, A046643, A046644, A165825, A257098, A298971, A299119, A299149 (numerators), A299151, A317848, A317929, A317932, A318314, A318512, A318653, A318681.

Cf. A318440 (the 2-adic valuation).

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
f[p_, e_] := 2^((1 + Mod[p, 2])*e - DigitCount[e, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n)={my(v=factor(n)[,2]); denominator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018

A299150(n) = { my(f = factor(n), m=1); for(i=1, #f~, m *= 2^(((1+(f[i,1]%2))*f[i,2]) - hammingweight(f[i,2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 08 2025

a(n) = denominator(n*A317848(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)). - Andrew Howroyd, Aug 09 2018
a(n) = A046644(n)/A006519(n). - Andrew Howroyd and Antti Karttunen, Aug 30 2018
From Antti Karttunen, Sep 03 2018: (Start)
a(n) = 2^A318440(n).
Multiplicative with a(2^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for odd primes p.
Multiplicative with a(p^e) = 2^(((1+A000035(p))*e)-A000120(e)) for all primes p.
(End)

Keyword:mult added by Andrew Howroyd, Aug 09 2018

A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 5, 27, 5, 11, 9, 13, 7, 15, 35, 17, 27, 19, 15, 21, 11, 23, 15, 75, 13, 135, 21, 29, 15, 31, 63, 33, 17, 35, 81, 37, 19, 39, 25, 41, 21, 43, 33, 135, 23, 47, 105, 147, 75, 51, 39, 53, 135, 55, 35, 57, 29, 59, 45, 61, 31, 189, 231, 65, 33

Offset: 1

Gus Wiseman, Feb 03 2018

Dirichlet convolution of a(n)/A046644(n) with itself yields A000265. - Antti Karttunen, Aug 30 2018

			Sequence begins: 1, 1, 3/2, 3/2, 5/2, 3/2, 7/2, 5/2, 27/8, 5/2, 11/2, 9/4, 13/2, 7/2.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms Andrew Howroyd)
Vaclav Kotesovec, Graph - the asymptotic ratio (65537 terms)
Wikipedia, Dirichlet convolution.

Cf. A000010, A000265, A003958, A007431, A018804, A023900, A029935, A046643, A046644, A165825, A257098, A298971, A299119, A299150 (denominators), A299151, A317848, A318319, A318321, A318649.

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
odd[n_] := n/2^IntegerExponent[n, 2]; f[p_, e_] := odd[p^e*Binomial[2*e, e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n)={my(v=factor(n)[,2]); numerator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dAndrew Howroyd, Aug 09 2018

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator(n*A317848(n)/A165825(n)) = A000265(n*A317848(n)). - Andrew Howroyd, Aug 09 2018

Sum_{k=1..n} A299149(k)/A299150(k) ~ n^2 / (2*sqrt(Pi*log(n))) * (1 + (1-gamma) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

Keyword:mult added by Andrew Howroyd, Aug 09 2018

A165826 Totally multiplicative sequence with a(p) = 5.

Original entry on oeis.org

1, 5, 5, 25, 5, 25, 5, 125, 25, 25, 5, 125, 5, 25, 25, 625, 5, 125, 5, 125, 25, 25, 5, 625, 25, 25, 125, 125, 5, 125, 5, 3125, 25, 25, 25, 625, 5, 25, 25, 625, 5, 125, 5, 125, 125, 25, 5, 3125, 25, 125

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000351, A001222, A061142, A165824, A165825.

5^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 28 2023

a(n) = A000351(A001222(n)) = 5^bigomega(n) = 5^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 5 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

A166634 Totally multiplicative sequence with a(p) = 4*(p-1) for prime p.

Original entry on oeis.org

1, 4, 8, 16, 16, 32, 24, 64, 64, 64, 40, 128, 48, 96, 128, 256, 64, 256, 72, 256, 192, 160, 88, 512, 256, 192, 512, 384, 112, 512, 120, 1024, 320, 256, 384, 1024, 144, 288, 384, 1024, 160, 768, 168, 640, 1024, 352, 184, 2048, 576, 1024

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A003958, A165825, A166633.

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]*4^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (4*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = 4*(f[k,1]-1)); factorback(f);} \\ Michel Marcus, May 20 2016

Multiplicative with a(p^e) = (4*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)-1))^e(k).

a(n) = A165825(n) * A003958(n) = 4^bigomega(n) * A003958(n) = 4^A001222(n) * A003958(n).

A317848 Multiplicative with a(p^e) = binomial(2*e, e).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 20, 6, 4, 2, 12, 2, 4, 4, 70, 2, 12, 2, 12, 4, 4, 2, 40, 6, 4, 20, 12, 2, 8, 2, 252, 4, 4, 4, 36, 2, 4, 4, 40, 2, 8, 2, 12, 12, 4, 2, 140, 6, 12, 4, 12, 2, 40, 4, 40, 4, 4, 2, 24, 2, 4, 12, 924, 4, 8, 2, 12, 4, 8, 2, 120, 2, 4, 12, 12, 4, 8, 2, 140

Offset: 1

Andrew Howroyd, Aug 08 2018

The Dirichlet convolution square of this sequence is A165825.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A000984, A006519, A037445, A046643, A046644, A165825, A299149, A299150.

f[p_, e_] := Binomial[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n)={my(v=factor(n)[,2]); prod(i=1, #v, binomial(2*v[i], v[i]))}

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d

A317848(n) = factorback(apply(e -> binomial(e+e,e),factor(n)[,2])); \\ Antti Karttunen, Sep 17 2018

A037445(n) = A006519(a(n)).
A046643(n) = numerator(a(n)/A165825(n)) = A000265(a(n)).
A046644(n) = denominator(a(n)/A165825(n)) = A165825(n)/A037445(n).
A299149(n) = numerator(n*a(n)/A165825(n)) = A000265(n*a(n)).
A299150(n) = denominator(n*a(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)).

A166644 Totally multiplicative sequence with a(p) = 4*(p+1) for prime p.

Original entry on oeis.org

1, 12, 16, 144, 24, 192, 32, 1728, 256, 288, 48, 2304, 56, 384, 384, 20736, 72, 3072, 80, 3456, 512, 576, 96, 27648, 576, 672, 4096, 4608, 120, 4608, 128, 248832, 768, 864, 768, 36864, 152, 960, 896, 41472, 168, 6144, 176, 6912, 6144, 1152, 192, 331776

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A003959, A165825.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*4^(PrimeOmega[n]), {n, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (4*(p+1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 17 2023 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = 4*(f[k,1]+1)); factorback(f);} \\ Michel Marcus, May 21 2016

Multiplicative with a(p^e) = (4*(p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (4*(p(k)+1))^e(k).

a(n) = A165825(n) * A003959(n) = 4^bigomega(n) * A003959(n) = 4^A001222(n) * A003959(n).

A351521 Dirichlet g.f.: Product_{p prime} (1 + 4*p^(-s)).

Original entry on oeis.org

1, 4, 4, 0, 4, 16, 4, 0, 0, 16, 4, 0, 4, 16, 16, 0, 4, 0, 4, 0, 16, 16, 4, 0, 0, 16, 0, 0, 4, 64, 4, 0, 16, 16, 16, 0, 4, 16, 16, 0, 4, 64, 4, 0, 0, 16, 4, 0, 0, 0, 16, 0, 4, 0, 16, 0, 16, 16, 4, 0, 4, 16, 0, 0, 16, 64, 4, 0, 16, 64, 4, 0, 4, 16, 0, 0, 16, 64

Offset: 1

Vaclav Kotesovec, Feb 17 2022

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A074823, A347149.
Cf. A061142, A165824, A165825.
Cf. A008966, A035116.

Table[MoebiusMu[n]^2 * 4^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X))[n], ", "))

Dirichlet g.f.: zeta(s)^4 * Product_{prime p} (1 + (4 - 15*p^s + 20*p^(2*s) - 10*p^(3*s))/p^(5*s)).
a(n) = A008966(n) * A035116(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 4, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

A166625 Totally multiplicative sequence with a(p) = 4p for prime p.

Original entry on oeis.org

1, 8, 12, 64, 20, 96, 28, 512, 144, 160, 44, 768, 52, 224, 240, 4096, 68, 1152, 76, 1280, 336, 352, 92, 6144, 400, 416, 1728, 1792, 116, 1920, 124, 32768, 528, 544, 560, 9216, 148, 608, 624, 10240, 164, 2688, 172, 2816, 2880, 736, 188, 49152, 784, 3200

Offset: 1

Jaroslav Krizek, Oct 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001222, A165825.

Table[n 4^PrimeOmega[n],{n,50}] (* Harvey P. Dale, Jan 20 2014 *)

a(n) = n*4^bigomega(n); \\ Altug Alkan, May 19 2016

Multiplicative with a(p^e) = (4p)^e.
If n = Product p(k)^e(k) then a(n) = Product (4*p(k))^e(k).
a(n) = n * A165825(n) = n * 4^bigomega(n) = n * 4^A001222(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 4 * p^(1 - s)). - Ilya Gutkovskiy, Oct 30 2019

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

Jaroslav Krizek, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001222, A165825, A166586.

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 *)

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).

cp's OEIS Frontend

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

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A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

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A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

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A165826 Totally multiplicative sequence with a(p) = 5.

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A166634 Totally multiplicative sequence with a(p) = 4*(p-1) for prime p.

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A317848 Multiplicative with a(p^e) = binomial(2*e, e).

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A166644 Totally multiplicative sequence with a(p) = 4*(p+1) for prime p.

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