A307607 -id:A307607 - cp's OEIS frontend

A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

1, 1, 3, 6, 17, 28, 72, 122, 282, 493, 1027, 1790, 3673, 6300, 12205, 21117, 39782, 67989, 124937, 212189, 381705, 644625, 1136315, 1905352, 3312916, 5513005, 9443362, 15624026, 26445046, 43451200, 72751824, 118792691, 196966722, 319714816, 525316191, 847734183, 1381904765

Offset: 0

Ilya Gutkovskiy, Apr 18 2019

nonn

Euler transform of A050369.

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 17*x^4 + 28*x^5 + 72*x^6 + 122*x^7 + 282*x^8 + 493*x^9 + ...

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

Cf. A050369, A074206, A129374, A307605, A307606, A307607.

terms = 36; A[] = 1; Do[A[x] = 1/(1 - x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*A074206(k)).

a(n) ~ (-Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(50*(2+r))) * exp(12/625 + ((2+r)/(1+r)) * (-Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (A^(144/625) * sqrt(2*Pi*(2+r)) * n^(1/2 + 1/(50*(2+r)))), where r = A107311 is the root of the equation zeta(r)=2 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 18 2021

A197953 a(n) = 1 + Sum_{d|n, d>1} d * a(n/d).

Original entry on oeis.org

1, 3, 4, 11, 6, 24, 8, 43, 22, 38, 12, 128, 14, 52, 54, 171, 18, 186, 20, 206, 74, 80, 24, 640, 56, 94, 130, 284, 30, 494, 32, 683, 114, 122, 118, 1226, 38, 136, 134, 1038, 42, 682, 44, 440, 432, 164, 48, 3072, 106, 488, 174, 518, 54, 1374, 182, 1436, 194

Offset: 1

Paul D. Hanna, Oct 19 2011

nonn

Logarithmic derivative of A129374, where g.f. G(x) of A129374 satisfies: G(x) = 1/(1-x) * G(x^2)*G(x^3)*G(x^4)*...*G(x^n)*...

			L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 24*x^6/6 +...
where
L(x) = -log(1-x) + L(x^2) + L(x^3) + L(x^4) + L(x^5) +...+ L(x^n) +...
also, exp(L(x)) is the g.f. of A129374:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 15*x^6 + 20*x^7 +...

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul D. Hanna)

Cf. A067824, A129374, A307607.

PARI
```
{a(n)=sumdiv(n,d,d*if(d==1,1,a(n/d)))}
```

/* L.g.f. satisfies: L(x) = -log(1-x) + Sum_{n>1} L(x^n) */
{a(n)=local(L=x,X=x+x*O(x^n));for(i=1,n,L=-log(1-X)+sum(m=2,n,subst(L,x,x^m+x*O(x^n))));n*polcoeff(L,n)}

L.g.f. satisfies: L(x) = -log(1-x) + Sum_{n>1} L(x^n), where L(x) = Sum_{n>=1} a(n)*x^n/n.

cp's OEIS Frontend

A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...

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A197953 a(n) = 1 + Sum_{d|n, d>1} d * a(n/d).

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cp's OEIS Frontend

A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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Author

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Crossrefs

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Mathematica

Formula

A197953 a(n) = 1 + Sum_{d|n, d>1} d * a(n/d).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A307604 G.f. A(x) satisfies: A(x) = (1/(1 - x)) * A(x^2)^2A(x^3)^3A(x^4)^4* ... A(x^k)^k ...