A299069 -id:A299069 - cp's OEIS frontend

A061255 Euler transform of Euler totient function phi(n), cf. A000010.

1, 1, 2, 4, 7, 13, 21, 37, 60, 98, 157, 251, 392, 612, 943, 1439, 2187, 3293, 4930, 7330, 10839, 15935, 23315, 33933, 49170, 70914, 101861, 145713, 207638, 294796, 417061, 588019, 826351, 1157651, 1616849, 2251623, 3126775, 4330271, 5981190

Offset: 0

Vladeta Jovovic, Apr 21 2001

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Alessandro De Luca, Gabriele Fici, and Andrea Frosini, Digital Convexity and Combinatorics on Words, arXiv:2502.19926 [math.CO], 2025. See p. 11.
N. J. A. Sloane, Transforms

Cf. A000010, A057660, A006171, A001970, A061256, A061257, A299069, A318975.

nn = 20; b = Table[EulerPhi[n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

G.f.: Product_{k>=1} (1 - x^k)^(-phi(k)).
a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k) = Sum_{d|k} d*phi(d), cf. A057660.
Logarithmic derivative yields A057660 (equivalent to above formula). - Paul D. Hanna, Sep 05 2012
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - 1/6) * A^2 * Zeta(3)^(1/9) / (2^(4/9) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} (sigma_2(k^2)/sigma_1(k^2)) * x^k/k). - Ilya Gutkovskiy, Apr 22 2019

A301876 Expansion of Product_{k>=1} (1 + x^k)^A007434(k).

Original entry on oeis.org

1, 1, 3, 11, 23, 63, 137, 329, 738, 1618, 3562, 7578, 16116, 33540, 69384, 141608, 286493, 574173, 1140355, 2247835, 4394415, 8532983, 16450061, 31513869, 59991541, 113536117, 213659967, 399910311, 744672519, 1379758479, 2544367633

Offset: 0

Vaclav Kotesovec, Mar 28 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A007434, A299069, A301875.

nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^j * Sum[Sum[d^2 MoebiusMu[k/d], {d, Divisors @ k}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp(2^(5/4) * Pi / 3^(5/4) * (7/(5*Zeta(3)))^(1/4) * n^(3/4)) *(7/(15*Zeta(3)))^(1/8) / (2^(15/8) * n^(5/8)).

A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 4, 10, 20, 42, 80, 154, 288, 522, 940, 1658, 2892, 4970, 8456, 14218, 23696, 39122, 64044, 104042, 167732, 268602, 427248, 675482, 1061632, 1659298, 2579676, 3990418, 6142892, 9412906, 14360136, 21814698, 33004704, 49739426, 74677924, 111713658

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A299069 and A061255.

			a(n) ~ exp(3^(4/3) * (7*Zeta(3))^(1/3) * n^(2/3) / (2*Pi^(2/3)) - 1/6) * A^2 * (7*Zeta(3))^(1/9) / (sqrt(2) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A061255, A299069, A301554, A301555.

Cf. A318976, A318814, A318977.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]

A326305 Dirichlet g.f.: zeta(s-1) * (1 - 2^(-s)) / zeta(s).

Original entry on oeis.org

1, 0, 2, 1, 4, 0, 6, 2, 6, 0, 10, 2, 12, 0, 8, 4, 16, 0, 18, 4, 12, 0, 22, 4, 20, 0, 18, 6, 28, 0, 30, 8, 20, 0, 24, 6, 36, 0, 24, 8, 40, 0, 42, 10, 24, 0, 46, 8, 42, 0, 32, 12, 52, 0, 40, 12, 36, 0, 58, 8, 60, 0, 36, 16, 48, 0, 66, 16, 44, 0, 70, 12, 72, 0, 40

Offset: 1

Ilya Gutkovskiy, Oct 17 2019

Moebius transform of A026741.
Dirichlet convolution of A002131 with Dirichlet inverse of A000005.
Dirichlet convolution of A000027 with Dirichlet inverse of A001511.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000027, A001511, A002131, A007427, A008683, A016825 (positions of 0's), A026741, A092673, A299069.

[IsOdd(n) select EulerPhi(n) else EulerPhi(n)-EulerPhi(n div 2) : n in [1..80]]; // Marius A. Burtea, Oct 17 2019

Table[Sum[MoebiusMu[n/d] Numerator[d/2], {d, Divisors[n]}], {n, 1, 75}]
a[n_] := If[OddQ[n], EulerPhi[n], EulerPhi[n] - EulerPhi[n/2]]; Table[a[n], {n, 1, 75}]
f[2, e_] := If[e == 1, 0, 2^(e - 2)]; f[p_, e_] := (p - 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = phi(n) if n odd, phi(n) - phi(n/2) if n even, where phi = A000010.
a(n) = Sum_{d|n} mu(n/d) * A026741(d).
a(n) = Sum_{d|n} A007427(n/d) * A002131(d).
a(n) = Sum_{d|n} A092673(n/d) * d.
a(p) = p - 1, where p is odd prime.
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A299069.
Sum_{k=1..n} a(k) ~ 9*n^2 / (4*Pi^2). - Vaclav Kotesovec, Oct 26 2019
Multiplicative with a(2^e) = 0 if e = 1 and 2^(e-2) otherwise, and a(p^e) = (p-1)*p^(e-1) for odd primes p. - Amiram Eldar, Nov 30 2020

A078747 Expansion of Sum_{k>0} kphi(k)x^k/(1+x^k).

Original entry on oeis.org

1, 1, 7, 5, 21, 7, 43, 21, 61, 21, 111, 35, 157, 43, 147, 85, 273, 61, 343, 105, 301, 111, 507, 147, 521, 157, 547, 215, 813, 147, 931, 341, 777, 273, 903, 305, 1333, 343, 1099, 441, 1641, 301, 1807, 555, 1281, 507, 2163, 595, 2101, 521, 1911, 785, 2757, 547

Offset: 1

Vladeta Jovovic, Dec 22 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A057660, A240976, A299069.

f[p_, e_] := If[p == 2, (4^e - 1)/3, (p^(2*e + 1) + 1)/(p + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 15 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, (4^f[i,2]-1)/3, (f[i,1]^(2*f[i,2]+1)+1)/(f[i,1]+1))); } \\ Amiram Eldar, Oct 15 2022

Multiplicative with a(2^e) = (4^e-1)/3, a(p^e) = (p^(2*e+1)+1)/(p+1), p>2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(4*zeta(2)) = 0.182690... (A240976). - Amiram Eldar, Oct 15 2022
Dirichlet g.f.: (zeta(s)*zeta(s-2)/zeta(s-1))*(1-2^(1-s)). - Amiram Eldar, Dec 30 2022

A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

Original entry on oeis.org

1, 1, 3, 9, 20, 52, 105, 253, 536, 1142, 2421, 4999, 10278, 20686, 41512, 81984, 161029, 312681, 603070, 1153284, 2189331, 4129537, 7733317, 14399693, 26644337, 49034811, 89741600, 163411148, 296074694, 533909026, 958416113, 1712893825, 3048468607, 5403248469, 9539609984

Offset: 0

Paul D. Hanna, May 26 2013

nonn

Here phi(n) = A000010(n) is the Euler totient function.

Euler transform of A002618. - Vaclav Kotesovec, Mar 30 2018

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 37*x^4/4 + 101*x^5/5 + 95*x^6/6 + 295*x^7/7 + 293*x^8/8 + 505*x^9/9 +...+ A068963(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A068963, A002618, A061255, A299069, A301986.

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m,d,eulerphi(d^3))*x^m/m)+x*O(x^n)),n)}
for(n=0,35,print1(a(n),", "))

a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - Vaclav Kotesovec, Mar 30 2018

A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A061255, A061256, A006171.
Cf. A299069, A192065, A107742.
Cf. A294361, A294363.

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

A301986 Expansion of Product_{k>=1} (1 + x^k)^(k*A000010(k)), where A000010 is the Euler totient function.

Original entry on oeis.org

1, 1, 2, 8, 15, 41, 75, 179, 378, 748, 1591, 3133, 6369, 12357, 24225, 46691, 89301, 169589, 318413, 596255, 1103468, 2036880, 3725353, 6786021, 12281026, 22107132, 39604155, 70566697, 125209095, 221048851, 388705826, 680465440, 1186649341, 2061086935

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A002618, A061255, A226106, A299069.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(j + 1)/j * Sum[k*EulerPhi[k] * x^(j*k), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]

a(n) ~ exp(2^(3/2) * 7^(1/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4))) * 7^(1/8) / (2^(7/4) * 5^(1/8) * Pi^(1/4) * n^(5/8)).

A328774 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} phi(n) * x^n, where phi = A000010.

Original entry on oeis.org

1, 1, 1, 1, 2, -2, 4, -3, 4, -7, 14, -21, 30, -38, 50, -79, 128, -190, 286, -419, 598, -895, 1386, -2121, 3178, -4733, 7122, -10796, 16414, -25011, 38056, -57722, 87568, -133308, 203618, -311318, 475536, -726069, 1109718, -1698185, 2601166, -3987305, 6114666, -9378656, 14389676

Offset: 1

Ilya Gutkovskiy, Oct 27 2019

sign

Inverse weigh transform of A000010.

Cf. A000010, A299069, A320778.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; numtheory[phi](n)-b(n, n-1) end:
seq(a(n), n=1..45);  # Alois P. Heinz, Oct 27 2019

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = EulerPhi[n] - b[n, n - 1]; Array[a, 45]

A321265 a(n) = [x^n] Product_{k>=1} (1 + x^k)^J_n(k), where J_() is the Jordan function.

Original entry on oeis.org

1, 1, 3, 33, 425, 12083, 665707, 68834806, 13654633905, 5535319947544, 4371956013518511, 6700051541666225780, 21029477920140943174285, 131152064162504305814647983, 1603485136950993248524876767297, 40291404321882574322412345562762188, 2031269423141309839019651314585293713041

Offset: 0

Ilya Gutkovskiy, Nov 01 2018

nonn

Wikipedia, Jordan's totient function

Cf. A059379, A059380, A299069, A301876, A321042, A321264.

Table[SeriesCoefficient[Product[(1 + x^k)^Sum[d^n MoebiusMu[k/d], {d, Divisors[k]}], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[Sum[Sum[(-1)^(k/d + 1) d j^n MoebiusMu[d/j], {j, Divisors[d]}], {d, Divisors[k]}] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

a(n) = [x^n] exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} (-1)^(k/d+1)*d*j^n*mu(d/j) ) * x^k/k).

cp's OEIS Frontend

A061255 Euler transform of Euler totient function phi(n), cf. A000010.

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A301876 Expansion of Product_{k>=1} (1 + x^k)^A007434(k).

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A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

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A326305 Dirichlet g.f.: zeta(s-1) * (1 - 2^(-s)) / zeta(s).

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Magma

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A078747 Expansion of Sum_{k>0} k*phi(k)*x^k/(1+x^k).

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A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

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A301986 Expansion of Product_{k>=1} (1 + x^k)^(k*A000010(k)), where A000010 is the Euler totient function.

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A078747 Expansion of Sum_{k>0} kphi(k)x^k/(1+x^k).