A165824 -id:A165824 - 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

A165825 Totally multiplicative sequence with a(p) = 4.

Original entry on oeis.org

1, 4, 4, 16, 4, 16, 4, 64, 16, 16, 4, 64, 4, 16, 16, 256, 4, 64, 4, 64, 16, 16, 4, 256, 16, 16, 64, 64, 4, 64, 4, 1024, 16, 16, 16, 256, 4, 16, 16, 256, 4, 64, 4, 64, 64, 16, 4, 1024, 16, 64

Offset: 1

Jaroslav Krizek, Sep 28 2009

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

Cf. A000302, A001222, A061142, A165824, A351521.

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

a(n) = 4^bigomega(n); \\ Altug Alkan, Apr 09 2016

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 4*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 17 2022

a(n) = A000302(A001222(n)) = 4^bigomega(n) = 4^A001222(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 4 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019
Sum_{k=1..n} a(k) = c * n^2 / (2 * log(2)) + O(n * log(n)^3), where c = Product_{p prime > 2} 1 / (1 - 4/p^2) = 2.6413142332392629671869467536904049315527375203817456105081927074458279809... - Vaclav Kotesovec, Feb 17 2022

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

A347149 Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

Original entry on oeis.org

1, 3, 3, 0, 3, 9, 3, 0, 0, 9, 3, 0, 3, 9, 9, 0, 3, 0, 3, 0, 9, 9, 3, 0, 0, 9, 0, 0, 3, 27, 3, 0, 9, 9, 9, 0, 3, 9, 9, 0, 3, 27, 3, 0, 0, 9, 3, 0, 0, 0, 9, 0, 3, 0, 9, 0, 9, 9, 3, 0, 3, 9, 0, 0, 9, 27, 3, 0, 9, 27, 3, 0, 3, 9, 0, 0, 9, 27, 3, 0, 0, 9, 3, 0, 9, 9, 9, 0, 3, 0, 9, 0, 9, 9, 9, 0, 3, 0, 0, 0

Offset: 1

Vaclav Kotesovec, Aug 20 2021

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Cf. A008966, A074823, A165824.
Cf. A013928, A069201, A048691.
Cf. A001620, A082633.

Table[MoebiusMu[n]^2 * 3^PrimeNu[n], {n, 1, 100}]

for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X))[n], ", "))

for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^3)[n], ", "))

Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)), then Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where f(1) = Product_{primes p} (1 - 6/p^2 + 8/p^3 - 3/p^4) = 0.1148840440802287887292512767015990978487135526872830176248484270625666728..., f'(1) = f(1) * Sum_{primes p} 12*log(p) / ((p-1)*(p+3)) = 0.5497153490016133577871571904347511299324572220423331992393596243955677299..., f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-24*p*(p-1) * log(p)^2 / ((p-1)^2 * (p+3)^2)) = 0.9028322988288094236586622799305270026576436536391185119652318723470259904... and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633).
a(n) = A008966(n) * A048691(n). - Enrique Pérez Herrero, Oct 27 2022
Multiplicative with a(p) = 3, and a(p^e) = 0 for e >= 2. - Amiram Eldar, Dec 25 2022

A166633 Totally multiplicative sequence with a(p) = 3*(p-1) for prime p.

Original entry on oeis.org

1, 3, 6, 9, 12, 18, 18, 27, 36, 36, 30, 54, 36, 54, 72, 81, 48, 108, 54, 108, 108, 90, 66, 162, 144, 108, 216, 162, 84, 216, 90, 243, 180, 144, 216, 324, 108, 162, 216, 324, 120, 324, 126, 270, 432, 198, 138, 486, 324, 432

Offset: 1

Jaroslav Krizek, Oct 18 2009

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

Cf. A001222, A003958, A165824, A166634.

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]*3^(PrimeOmega[m]), {m, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (3*(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] = 3*(f[k,1]-1)); factorback(f);} \\ Michel Marcus, May 20 2016

Multiplicative with a(p^e) = (3*(p-1))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)-1))^e(k).

a(n) = A165824(n) * A003958(n) = 3^bigomega(n) * A003958(n) = 3^A001222(n) * A003958(n).

A350961 a(n) = Sum_{k=1..n} 3^Omega(k).

Original entry on oeis.org

1, 4, 7, 16, 19, 28, 31, 58, 67, 76, 79, 106, 109, 118, 127, 208, 211, 238, 241, 268, 277, 286, 289, 370, 379, 388, 415, 442, 445, 472, 475, 718, 727, 736, 745, 826, 829, 838, 847, 928, 931, 958, 961, 988, 1015, 1024, 1027, 1270, 1279, 1306, 1315, 1342, 1345, 1426, 1435, 1516, 1525, 1534, 1537, 1618

Offset: 1

N. J. A. Sloane, Feb 06 2022

Tenenbaum, G. (2015). Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. See page 59.

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

Cf. A001222 (Omega), A069205, A069212. Partial sums of A165824.

Accumulate[3^PrimeOmega[Range[100]]] (* Vaclav Kotesovec, Feb 16 2022 *)

from sympy.ntheory.factor_ import primeomega
def A350961(n): return sum(3**primeomega(m) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

A166643 Totally multiplicative sequence with a(p) = 3*(p+1) for prime p.

Original entry on oeis.org

1, 9, 12, 81, 18, 108, 24, 729, 144, 162, 36, 972, 42, 216, 216, 6561, 54, 1296, 60, 1458, 288, 324, 72, 8748, 324, 378, 1728, 1944, 90, 1944, 96, 59049, 432, 486, 432, 11664, 114, 540, 504, 13122, 126, 2592, 132, 2916, 2592, 648, 144, 78732, 576, 2916

Offset: 1

Jaroslav Krizek, Oct 18 2009

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

Cf. A001222, A003959, A165824.

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*3^(PrimeOmega[n]), {n, 1, 100}] (* G. C. Greubel, May 20 2016 *)
f[p_, e_] := (3*(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] = 3*(f[k,1]+1)); factorback(f);} \\ Michel Marcus, May 21 2016

Multiplicative with a(p^e) = (3*(p+1))^e. If n = Product p(k)^e(k) then a(n) = Product (3*(p(k)+1))^e(k).

a(n) = A165824(n) * A003959(n) = 3^bigomega(n) * A003959(n) = 3^A001222(n) * A003959(n).

A349922 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 3 * k^(-s)).

Original entry on oeis.org

1, 3, 3, 12, 3, 12, 3, 39, 12, 12, 3, 48, 3, 12, 12, 129, 3, 48, 3, 48, 12, 12, 3, 165, 12, 12, 39, 48, 3, 57, 3, 399, 12, 12, 12, 201, 3, 12, 12, 165, 3, 57, 3, 48, 48, 12, 3, 552, 12, 48, 12, 48, 3, 165, 12, 165, 12, 12, 3, 237, 3, 12, 48, 1245, 12, 57, 3, 48, 12, 57

Offset: 1

Ilya Gutkovskiy, Dec 05 2021

nonn

Cf. A001055, A165824, A242587, A339318, A349921.

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

A166624 Totally multiplicative sequence with a(p) = 3p for prime p.

Original entry on oeis.org

1, 6, 9, 36, 15, 54, 21, 216, 81, 90, 33, 324, 39, 126, 135, 1296, 51, 486, 57, 540, 189, 198, 69, 1944, 225, 234, 729, 756, 87, 810, 93, 7776, 297, 306, 315, 2916, 111, 342, 351, 3240, 123, 1134, 129, 1188, 1215, 414, 141, 11664, 441, 1350

Offset: 1

Jaroslav Krizek, Oct 18 2009

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

Cf. A001222, A165824.

Table[n*3^PrimeOmega[n], {n, 1, 100}] (* G. C. Greubel, May 19 2016 *)

Multiplicative with a(p^e) = (3p)^e.
If n = Product p(k)^e(k) then a(n) = Product (3*p(k))^e(k).
a(n) = n * A165824(n) = n * 3^bigomega(n) = n * 3^A001222(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 3 * p^(1 - s)). - Ilya Gutkovskiy, Oct 30 2019

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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A165825 Totally multiplicative sequence with a(p) = 4.

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

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A347149 Dirichlet g.f.: Product_{primes p} (1 + 3/p^s).

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A166633 Totally multiplicative sequence with a(p) = 3*(p-1) for prime p.

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A350961 a(n) = Sum_{k=1..n} 3^Omega(k).

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A166643 Totally multiplicative sequence with a(p) = 3*(p+1) for prime p.

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A349922 Dirichlet g.f.: Product_{k>=2} 1 / (1 - 3 * k^(-s)).

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