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 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane, Dec 08 2001

Keywords

Comments

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

References

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

Links

Crossrefs

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).

Programs

  • Maple
    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;
  • Mathematica
    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 *)
  • PARI
    for(n=1,100,print1(n*sumdiv(n,d,moebius(d)^2/d^2),","))
  • PARI
    a(n)=sumdiv(n,d,moebius(n/d)^2*d^2); /* Joerg Arndt, Jul 06 2011 */

Formula

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

A156733 Euler transform of n*A065958(n).

Original entry on oeis.org

1, 1, 11, 41, 176, 606, 2391, 8091, 28636, 95056, 316048, 1014240, 3237325, 10082015, 31109500, 94352346, 283209381, 838650191, 2458835711, 7127912979, 20471486368, 58224189612, 164181018330, 458982667630, 1273039111210, 3503609456548, 9572771822745, 25971150308985
Offset: 0

Views

Author

Paul D. Hanna and Vladeta Jovovic, Feb 14 2009

Keywords

Comments

Compare to the g.f. of planar partitions (A000219): exp( Sum_{n>=1} sigma(n,2)*x^n/n ) = Product_{n>=1} 1/(1-x^n)^n.

Links

Crossrefs

Cf. A001157, A156303, A065827, A301978, A301980.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*numtheory[sigma][2](j^2), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 24 2016
  • Mathematica
    a[0] = 1;
a[n_] := a[n] = (1/n) Sum[DivisorSigma[2, k^2] a[n-k], {k, 1, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 03 2020 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,sigma(m^2,2)*x^m/m)+x*O(x^n)),n)}
for(n=0,21,print1(a(n),", "))

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma_2(k^2)*a(n-k) for n>0, with a(0) = 1.
G.f.: exp( Sum_{n>=1} A065827(n)*x^n/n ), where A065827(n) = sigma_2(n^2) is the sum of squares of the divisors of n^2. - Paul D. Hanna, Aug 09 2012

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

Views

Author

Vaclav Kotesovec, Mar 30 2018

Keywords

Links

Crossrefs

Cf. A065958, A156733, A301978.

Programs

  • Mathematica
    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 *)

Formula

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)).
Showing 1-3 of 3 results.

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

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