A156733 -id:A156733 - cp's OEIS frontend

A065958 a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).

1, 5, 10, 20, 26, 50, 50, 80, 90, 130, 122, 200, 170, 250, 260, 320, 290, 450, 362, 520, 500, 610, 530, 800, 650, 850, 810, 1000, 842, 1300, 962, 1280, 1220, 1450, 1300, 1800, 1370, 1810, 1700, 2080, 1682, 2500, 1850, 2440, 2340, 2650, 2210

Offset: 1

N. J. A. Sloane, Dec 08 2001

The sequence may be considered as psi_2, a generalization of Dedekind psi function, where psi_1 is A001615. - Enrique Pérez Herrero, Jul 06 2011

József Sándor, Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press, Rehoboth 2002, pp. 193.

E. Pérez Herrero, Table of n, a(n) for n = 1..10000
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 Chapter 3.14.2

Cf. A000010, A007434, A156733, A301978, A301980, A321973.

Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), this sequence (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

A065958 := proc(n) local i,j,k,t1,t2,t3; t1 := ifactors(n)[2]; t2 := n^2*mul((1+1/(t1[i][1])^2),i=1..nops(t1)); end;

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A065958[n_]:=JordanTotient[n,4]/JordanTotient[n,2]; (* Enrique Pérez Herrero, Aug 22 2010 *)
f[p_, e_] := p^(2*e) + p^(2*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

for(n=1,100,print1(n*sumdiv(n,d,moebius(d)^2/d^2),","))

a(n)=sumdiv(n,d,moebius(n/d)^2*d^2); /* Joerg Arndt, Jul 06 2011 */

Multiplicative with a(p^e) = p^(2*e) + p^(2*e-2). - Vladeta Jovovic, Dec 09 2001
a(n) = n^2 * Sum_{d|n} mu(d)^2/d^2 - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(d)^2*d^2. - Joerg Arndt, Jul 06 2011
Inverse Euler transform of n*A156733(n). - Paul D. Hanna and Vladeta Jovovic, Feb 14 2009
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_4(n)/(phi(n)*psi(n)) = A059377(n)/(A001615(n)*A000010(n))
a(n) = J_4(n)/J_2(n) = A059377(n)/A007434(n), where J_k is the k-th Jordan totient function. (End)
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s). Dirichlet convolution of A008966 and A000290. - R. J. Mathar, Apr 10 2011
G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^4 - 1)) = 1.5421162831401587416523241690601522041445615542162573163112157073779258386... - Vaclav Kotesovec, Sep 19 2020
a(n) = Sum_{d|n} d*phi(d)*psi(n/d). - Ridouane Oudra, Jan 01 2021
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} psi(gcd(n,k))*n/gcd(n,k), where psi(n) = A001615(n).
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^3, where c = 315*zeta(3)/Pi^6 = 0.393854... . - Amiram Eldar, Oct 19 2022

A301978 Euler transform of A065958.

Original entry on oeis.org

1, 1, 6, 16, 51, 127, 367, 897, 2342, 5662, 13894, 32656, 77076, 176586, 403526, 904140, 2013267, 4418167, 9628682, 20741434, 44362988, 93984842, 197731390, 412619250, 855408327, 1760687593, 3601827236, 7321181534, 14796204874, 29730215150, 59419375058

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

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

Cf. A065958, A156733, A301980.

nmax = 40; CoefficientList[Series[Exp[Sum[Sum[Sum[MoebiusMu[k/d]^2*d^2, {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 *)

G.f.: Product_{k>=1} 1/(1-x^k)^A065958(k).

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

A301980 Expansion of Product_{k>=1} (1 + x^k)^A065958(k).

Original entry on oeis.org

1, 1, 5, 15, 40, 106, 281, 685, 1690, 4050, 9496, 21908, 49902, 111740, 247465, 541353, 1171070, 2507602, 5319085, 11178947, 23298878, 48169708, 98834943, 201335651, 407345067, 818767703, 1635528657, 3247634227, 6412057831, 12590738729, 24593652845

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

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

Cf. A065958, A156733, A301978.

nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^j * Sum[Sum[MoebiusMu[k/d]^2*d^2, {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) * sqrt(7) * Zeta(3)^(1/4) * n^(3/4) / sqrt(3*Pi) - sqrt(Pi) * n^(1/4) / (2^(9/4) * 3^(3/2) * sqrt(7) * Zeta(3)^(1/4))) * 21^(1/4) * Zeta(3)^(1/8) / (2^(15/8) * Pi^(3/4) * n^(5/8)).

A381679 Euler transform of A000056.

Original entry on oeis.org

1, 1, 7, 31, 100, 364, 1152, 3864, 12102, 37358, 113618, 337562, 990798, 2857926, 8144334, 22902470, 63660695, 175026047, 476242001, 1283435153, 3427047146, 9072455146, 23820491998, 62057045134, 160471504373, 412022656517, 1050740365571, 2662223436203

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A061255, A381680.

Cf. A000056, A084218, A156733.

a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[4, k^2]/DivisorSigma[2, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)/sigma(k^2, 2)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^A000056(k).
G.f.: exp( Sum_{k>=1} sigma_4(k^2)/sigma_2(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2)/sigma_2(k^2) * a(n-k).
a(n) ~ exp(5*(3*zeta(5)/zeta(3))^(1/5) * n^(4/5) / 2^(7/5) - 1/10 - 12*zeta'(-3)) * A^(6/5) * (3*zeta(5)/zeta(3))^(3/25) / (2^(7/50) * sqrt(5*Pi) * n^(31/50)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 04 2025

A381170 Euler transform of n^2 * A065959(n).

Original entry on oeis.org

1, 1, 37, 289, 2107, 14329, 105187, 693579, 4512054, 28468770, 176428599, 1065826203, 6323626404, 36816785552, 210944620532, 1189766311028, 6615412814561, 36287015790029, 196547683500294, 1051919158699442, 5566679104757415, 29144209704259923, 151039019038054896

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A156733, A381709.

Cf. A065959.

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 3)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^(k^2 * A065959(k)).
G.f.: exp( Sum_{k>=1} sigma_3(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_3(k^2) * a(n-k).

A381709 Euler transform of n^3 * A065960(n).

Original entry on oeis.org

1, 1, 137, 2351, 29075, 408429, 5957562, 76590384, 955079422, 11831378688, 142650905559, 1668991927795, 19144774189917, 215790313316371, 2388025355854986, 25973791505651972, 278176027053878678, 2936495245822593766, 30573379794788083289, 314185573464039742503

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A156303, A156733, A381170.

Cf. A023872, A065960, A381679.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^(k^3 * A065960(k)).
G.f.: exp( Sum_{k>=1} sigma_4(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2) * a(n-k).

cp's OEIS Frontend

A065958 a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).

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A301978 Euler transform of A065958.

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A301980 Expansion of Product_{k>=1} (1 + x^k)^A065958(k).

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A381679 Euler transform of A000056.

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A381170 Euler transform of n^2 * A065959(n).

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A381709 Euler transform of n^3 * A065960(n).

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