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.

A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 10, 30, 57, 152, 289, 676, 1304, 2809, 5335, 10961, 20487, 40329, 74476, 141914, 258094, 479638, 860025, 1563716, 2767982, 4940567, 8636563, 15173805, 26217392, 45416811, 77629455, 132800937, 224695510, 380079521, 637006921
Offset: 0

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Author

Paul D. Hanna, Feb 08 2009

Keywords

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 57*x^5 + 152*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 6^2*x^5/5 + 12^2*x^6/6 +...
Also log(A(x)) = (x + 3*x^2 + 4*x^3 + 7*x^4 +...+ sigma(k)*x^k +...)/1 +
(3*x^2 + 7*x^4 + 12*x^6 + 15*x^8 + 18*x^10 +...+ sigma(2*k)*x^(2*k) +...)/2 +
(4*x^3 + 12*x^6 + 13*x^9 + 28*x^12 + 24*x^15 +...+ sigma(3*k)*x^(3*k) +...)/3 +
(7*x^4 + 15*x^8 + 28*x^12 + 31*x^16 + 42*x^20 +...+ sigma(4*k)*x^(4*k) +...)/4 +
(6*x^5 + 18*x^10 + 24*x^15 + 42*x^20 + 31*x^25 +...+ sigma(5*k)*x^(5*k) +...)/5 +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A072861, A178933, A205797, A382125.

Programs

  • Mathematica
    nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2024 *)
  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,sigma(k)^2*x^k/k)+x*O(x^n)),n)}
  • PARI
    {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^2*a(n-k)))}
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,sum(k=1,n\m,sigma(m*k)*x^(m*k)/m)+x*O(x^n))),n)}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^2*a(n-k) for n>0, with a(0) = 1.
Euler transform of A060648. [From Vladeta Jovovic, Feb 14 2009]
It appears that G.f.: A(x)=prod(n=1,infinity, E(x^n)^(-A001615(n))) where E(x) = prod(n=1,infinity,1-x^n). [From Joerg Arndt, Dec 30 2010]
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k) * x^(n*k) / n ). [From Paul D. Hanna, Jan 23 2012]
log(a(n)) ~ 3*(5*zeta(3))^(1/3) * n^(2/3) / 2. - Vaclav Kotesovec, Oct 29 2024

A382123 a(n) = sigma(n)*sigma(2*n)/3 for n >= 1.

Original entry on oeis.org

1, 7, 16, 35, 36, 112, 64, 155, 169, 252, 144, 560, 196, 448, 576, 651, 324, 1183, 400, 1260, 1024, 1008, 576, 2480, 961, 1372, 1600, 2240, 900, 4032, 1024, 2667, 2304, 2268, 2304, 5915, 1444, 2800, 3136, 5580, 1764, 7168, 1936, 5040, 6084, 4032, 2304, 10416, 3249, 6727, 5184, 6860
Offset: 1

Views

Author

Paul D. Hanna, Apr 06 2025

Keywords

Comments

For n >= 1, 2*A329963(n) = A087943(k) for some k; this is a consequence of the prime factorization properties of the numbers listed in A329963 and A087943 (see the comments in both entries). That is, two times any term found in A329963 (numbers k such that sigma(k) is not divisible by 3) equals a term found in A087943 (numbers k such that 3 divides sigma(k)). Therefore sigma(n)*sigma(2*n) is divisible by 3 for n >= 1.
Equals the logarithmic derivative of A382124.

Crossrefs

Cf. A382124, A382125, A087943, A329963, A347108.
Cf. A000203, A062731.

Programs

  • PARI
    {a(n) = sigma(n)*sigma(2*n)/3}
for(n=1,52, print1(a(n),", "))

Formula

a(n) = A000203(n) * A062731(n) / 3.
Sum_{k=1..n} a(k) ~ 2*zeta(3)*n^3/3. - Vaclav Kotesovec, Apr 06 2025

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

1, 1, 4, 9, 22, 44, 105, 200, 425, 825, 1634, 3072, 5917, 10846, 20153, 36436, 65882, 116831, 207293, 361502, 629539, 1083068, 1856251, 3150554, 5328137, 8933266, 14920357, 24745481, 40869317, 67089425, 109697089, 178379353, 288953043, 465805681, 748079686, 1196148976, 1905801579, 3024212984
Offset: 0

Views

Author

Paul D. Hanna, Apr 06 2025

Keywords

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Equals the self-convolution cube root of A382125.
Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + 3072*x^11 + 5917*x^12 + ...
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + ... + A382125(n)*x^n + ...

Links

Crossrefs

Cf. A382125, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).
Cf. A087943, A329963.

Programs

  • Mathematica
    nmax=37; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/(3n) ,{n,nmax}]],{x,0,nmax}],x] (* Stefano Spezia, Apr 06 2025 *)
  • PARI
    {a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)/3*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( (1/3) * Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( (1/3) * Sum_{n>=1} (1/n) * Sum_{k>=1} sigma(2*n*k) * x^(n*k) ).
(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k)/3 * a(n-k) for n > 0, with a(0) = 1.
Showing 1-3 of 3 results.

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