A198299 -id:A198299 - cp's OEIS frontend

A198296 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n) ).

1, 1, 2, 3, 6, 8, 17, 22, 44, 62, 115, 154, 311, 409, 754, 1070, 1949, 2639, 4917, 6645, 12055, 16916, 29594, 40719, 73907, 100959, 176010, 248207, 429626, 594220, 1040624, 1436936, 2473555, 3486360, 5901887, 8233872, 14174779, 19689223, 33203829, 46967767

Offset: 0

Paul D. Hanna, Jan 26 2012

nonn

Logarithmic derivative yields A198299.

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...
such that, by definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)) + (x^3/3)/((1-x^3)*(1-3*x^3)) + (x^4/4)/((1-x^4)*(1-2*x^4)*(1-4*x^4)) + (x^5/5)/((1-x^5)*(1-5*x^5)) + (x^6/6)/((1-x^6)*(1-2*x^6)*(1-3*x^6)*(1-6*x^6)) +...+ (x^n/n)/Product_{d|n} (1-d*x^n) +...
Also, we have the identity:
log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2
+ (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3
+ (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4
+ (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5
+ (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...
+ exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...
Explicitly, the logarithm begins:
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 + 8*x^7/7 + 83*x^8/8 + 49*x^9/9 + 178*x^10/10 +...+ A198299(n)*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A198299 (log), A203320, A185301, A198304.

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*exp(sum(k=1,n\m,sigma(m,k)*x^(m*k)/k)+x*O(x^n)))),n)}

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*exp(sumdiv(m,d,-log(1-d*x^m+x*O(x^n)))))),n)}

G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k ) ), where sigma(n,k) is the sum of the k-th powers of the divisors of n.

A198302 a(n) = Sum_{d|n} d * sigma(n/d, d).

Original entry on oeis.org

1, 5, 7, 21, 11, 65, 15, 133, 106, 245, 23, 1077, 27, 1041, 1637, 3365, 35, 9992, 39, 18361, 16401, 22841, 47, 134461, 15686, 106917, 179494, 355173, 59, 1220075, 63, 1593189, 1952705, 2228909, 631005, 13778268, 75, 9962313, 20732901, 34805473, 83, 113693883

Offset: 1

Paul D. Hanna, Jan 27 2012

Here sigma(n,k) is the sum of the k-th powers of the divisors of n.

Logarithmic derivative of A198301.

			L.g.f.: L(x) = x + 5*x^2/2 + 7*x^3/3 + 21*x^4/4 + 11*x^5/5 + 65*x^6/6 + ...
Exponentiation yields the g.f. of A198301: exp(L(x)) = 1 + x + 3*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 42*x^6 + 62*x^7 + 131*x^8 + 206*x^9 + 398*x^10 + ... + A198301(n)*x^n + ...

Robert Israel, Table of n, a(n) for n = 1..6236

Cf. A000203, A198301 (exp), A185301, A198299.

f:= proc(n) local d; add(d*numtheory:-sigma[d](n/d),d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Nov 05 2024

a[n_] := DivisorSum[n, # * DivisorSigma[#, n/#] &]; Array[a, 40] (* Amiram Eldar, Aug 18 2023 *)

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

{a(n)=n*polcoeff(sum(m=1,n,sum(k=1,n\m,sigma(m,k)*x^(m*k)/m)+x*O(x^n)),n)}

L.g.f.: Sum_{n>=1} Sum_{k>=1} sigma(n,k) * x^(n*k)/n.

A198305 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - n*x^d/d).

Original entry on oeis.org

1, 3, 7, 19, 51, 159, 519, 1867, 7234, 30243, 135125, 642307, 3231047, 17138845, 95554662, 558384955, 3411049542, 21730279218, 144048688538, 991665854999, 7077433997172, 52283785492733, 399238054300828, 3147127294177099, 25579801627862301, 214139186144996635

Offset: 1

Paul D. Hanna, Jan 27 2012

nonn

Forms the logarithmic derivative of A198304.

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 51*x^5/5 + 159*x^6/6 +...
such that, by definition:
L(x) = x/(1-x) + (x^2/2)/((1-2*x)*(1-x^2)) + (x^3/3)/((1-3*x)*(1-x^3)) + (x^4/4)/((1-4*x)*(1-2*x^2)*(1-x^4)) + (x^5/5)/((1-5*x)*(1-x^5)) + (x^6/6)/((1-6*x)*(1-3*x^2)*(1-2*x^3)*(1-x^6)) +...+ (x^n/n)/Product_{d|n} (1-n*x^d/d) +...
Exponentiation yields the g.f. of A198304:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 54*x^6 + 148*x^7 +...

Cf. A198304 (exp), A198299.

{a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -log(1-m*x^d/d+x*O(x^n))))), n)}

cp's OEIS Frontend

A198296 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A198302 a(n) = Sum_{d|n} d * sigma(n/d, d).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A198305 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - n*x^d/d).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI