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 1-7 of 7 results.

A205811 G.f.: Product_{n>=1} [ (1 - x^n) / (1 - (n+1)^n*x^n) ]^(1/n).

Original entry on oeis.org

1, 1, 6, 29, 221, 1897, 23502, 335334, 5923570, 119354491, 2758647259, 71079498533, 2031108928680, 63520842121792, 2161164726505952, 79394066773371245, 3133259427956392983, 132166451829847198316, 5934636812034634649249, 282609413111134846839482
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2012

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 29*x^3 + 221*x^4 + 1897*x^5 + 23502*x^6 +...
where the g.f. equals the product:
A(x) = (1-x)/(1-2*x) * ((1-x^2)/(1-3^2*x^2))^(1/2) * ((1-x^3)/(1-4^3*x^3))^(1/3) * ((1-x^4)/(1-5^4*x^4))^(1/4) * ((1-x^5)/(1-6^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A205812:
log(A(x)) = x + 11*x^2/2 + 70*x^3/3 + 719*x^4/4 + 7806*x^5/5 + 122534*x^6/6 +...

Crossrefs

Cf. A205812 (log), A205814, A023881.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=1,m,binomial(m,k)*sigma(m,k))+x*O(x^n))),n)}
  • PARI
    {a(n)=polcoeff(prod(k=1,n,((1-x^k)/(1-(k+1)^k*x^k +x*O(x^n)))^(1/k)),n)}

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) ).

A205815 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k).

Original entry on oeis.org

1, 17, 136, 1585, 16986, 282338, 4784900, 101750689, 2359918963, 62200943002, 1792160567088, 56765070059074, 1946195069937314, 72080471103535786, 2862427829603121696, 121449533922041845569, 5480386857784931063958, 262149577935595804303451
Offset: 1

Views

Author

Paul D. Hanna, Feb 01 2012

Keywords

Comments

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

Examples

			L.g.f.: L(x) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...
Exponentiation yields the g.f. of A205814:
exp(L(x)) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1)*2 + 1*sigma(2,2)*1 = 2*3*2 + 1*5*1 = 17;
a(3) = 3*sigma(3,1)*4 + 3*sigma(3,2)*2 + 1*sigma(3,3)*1 = 3*4*4 + 3*10*2 + 1*28*1 = 136;
a(4) = 4*sigma(4,1)*8 + 6*sigma(4,2)*4 + 4*sigma(4,3)*2 + 1*sigma(4,3)*1 = 4*7*8 + 6*21*4 + 4*73*2 + 1*273*1 = 1585.

Crossrefs

Cf. A205814 (exp), A205812.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * 2^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 08 2016 *)
  • PARI
    {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*2^(n-k))}

Formula

Logarithmic derivative of A205814.
a(n) = Sum_{d|n} ((d+2)^n - 2^n).
a(n) ~ exp(2) * n^n. - Vaclav Kotesovec, Oct 08 2016

A206813 Position of 3^n in joint ranking of {2^i}, {3^j}, {5^k}.

Original entry on oeis.org

2, 6, 9, 12, 15, 19, 22, 25, 29, 31, 35, 39, 41, 45, 48, 51, 54, 58, 61, 64, 68, 71, 74, 78, 81, 84, 87, 91, 93, 97, 101, 103, 107, 110, 113, 117, 120, 123, 126, 130, 132, 136, 140, 143, 146, 149, 153, 156, 159, 163, 165, 169, 173, 175, 179, 182, 185, 188
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2012

Keywords

Comments

The exponents i,j,k range through the set N of positive integers, so that the position sequences (A206812 for 2^n, A206813 for 3^n, A206814 for 5^n) partition N.

Examples

			The joint ranking begins with 2,3,4,5,8,9,16,25,27,32,64,81,125,128,243,256, so that
A205812=(1,3,5,7,10,11,14,...)
A205813=(2,6,9,12,15,...)
A205814=(4,8,13,18,23,...)

Crossrefs

Cf. A206805, A206812, A206814.

Programs

  • Mathematica
    f[1, n_] := 2^n; f[2, n_] := 3^n;
f[3, n_] := 5^n; z = 1000;
d[n_, b_, c_] := Floor[n*Log[b, c]];
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2], t[3]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z/8}]] (* A206812 *)
Table[n + d[n, 3, 2] + d[n, 5, 2],
  {n, 1, 50}]                        (* A206812 *)
Flatten[Table[p[2, n], {n, 1, z/8}]] (* A206813 *)
Table[n + d[n, 2, 3] + d[n, 5, 3],
  {n, 1, 50}]                        (* A206813 *)
Flatten[Table[p[3, n], {n, 1, z/8}]] (* A206814 *)
Table[n + d[n, 2, 5] + d[n, 3, 5],
  {n, 1, 50}]                        (* A206814 *)

Formula

A205812(n) = n + [n*log(base 3)(2)] + [n*log(base 5)(2)],
A205813(n) = n + [n*log(base 2)(3)] + [n*log(base 5)(3)],
A205814(n) = n + [n*log(base 2)(5)] + [n*log(base 3)(5)],
where []=floor.

A206814 Position of 5^n in joint ranking of {2^i}, {3^j}, {5^k}.

Original entry on oeis.org

4, 8, 13, 18, 23, 27, 33, 37, 42, 47, 52, 56, 62, 66, 70, 76, 80, 85, 90, 95, 99, 105, 109, 114, 119, 124, 128, 134, 138, 142, 147, 152, 157, 161, 167, 171, 176, 181, 186, 190, 196, 200, 204, 210, 214, 219, 224, 229, 233, 239, 243, 248, 253, 258, 262
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2012

Keywords

Comments

The exponents i,j,k range through the set N of positive integers, so that the position sequences (A206812 for 2^n, A206813 for 3^n, A206814 for 5^n) partition N.

Examples

			The joint ranking begins with 2,3,4,5,8,9,16,25,27,32,64,81,125,128,243,256, so that
A205812=(1,3,5,7,10,11,14,...)
A205813=(2,6,9,12,15,...)
A205814=(4,8,13,18,23,...)

Crossrefs

Cf. A206805, A206812, A206813.

Programs

  • Mathematica
    f[1, n_] := 2^n; f[2, n_] := 3^n;
f[3, n_] := 5^n; z = 1000;
d[n_, b_, c_] := Floor[n*Log[b, c]];
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2], t[3]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z/8}]] (* A206812 *)
Table[n + d[n, 3, 2] + d[n, 5, 2],
  {n, 1, 50}]                        (* A206812 *)
Flatten[Table[p[2, n], {n, 1, z/8}]] (* A206813 *)
Table[n + d[n, 2, 3] + d[n, 5, 3],
  {n, 1, 50}]                        (* A206813 *)
Flatten[Table[p[3, n], {n, 1, z/8}]] (* A206814 *)
Table[n + d[n, 2, 5] + d[n, 3, 5],
  {n, 1, 50}]                        (* A206814 *)

Formula

A205812(n) = n + [n*log(base 3)(2)] + [n*log(base 5)(2)],
A205813(n) = n + [n*log(base 2)(3)] + [n*log(base 5)(3)],
A205814(n) = n + [n*log(base 2)(5)] + [n*log(base 3)(5)],
where []=floor.

A206766 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 3^(n-k).

Original entry on oeis.org

1, 23, 226, 3039, 33306, 594902, 10012010, 220553599, 5170061143, 138942811678, 4049569009674, 130045043225838, 4503599691290714, 168477832912220134, 6746676272050878036, 288487396687082933759, 13107200000016921588858, 630907565930072760920429
Offset: 1

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

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

Examples

			L.g.f.: L(x) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...
Exponentiation yields the g.f. of A206765:
exp(L(x)) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...
Illustration of terms.
a(2) = 2*sigma(2,1)*3 + 1*sigma(2,2)*1 = 2*3*3 + 1*5*1 = 23;
a(3) = 3*sigma(3,1)*9 + 3*sigma(3,2)*3 + 1*sigma(3,3)*1 = 3*4*9 + 3*10*3 + 1*28*1 = 226;
a(4) = 4*sigma(4,1)*27 + 6*sigma(4,2)*9 + 4*sigma(4,3)*3 + 1*sigma(4,4)*1 = 4*7*27 + 6*21*9 + 4*73*3 + 1*273*1 = 3039.

Links

Crossrefs

Cf. A206765 (exp), A205815, A205812.

Programs

  • PARI
    {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*3^(n-k))}
  • PARI
    {a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-3^k*x^k)/(1-(k+3)^k*x^k +x*O(x^n)))), n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) = Sum_{d|n} ((d+3)^n - 3^n).
Logarithmic derivative of A206765.
L.g.f.: Sum_{n>=1} (1/n) * log( (1 - 3^n*x^n) / (1 - (n+3)^n*x^n) ).
a(n) ~ exp(3) * n^n. - Vaclav Kotesovec, Oct 04 2020

A206764 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * (-1)^(n-k).

Original entry on oeis.org

1, -1, 10, 79, 1026, 15686, 279938, 5771359, 134218243, 3487832974, 100000000002, 3138673052878, 106993205379074, 3937454749863382, 155568096631586820, 6568441588686506943, 295147905179352825858, 14063102470280932000757, 708235345355337676357634
Offset: 1

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

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

Examples

			L.g.f.: L(x) = x - x^2/2 + 10*x^3/3 + 79*x^4/4 + 1026*x^5/5 + 15686*x^6/6 +...
Exponentiation yields the g.f. of A206763:
exp(L(x)) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +...
Illustration of terms.
a(2) = -2*sigma(2,1) + 1*sigma(2,2) = -2*3 + 1*5 = -1;
a(3) = 3*sigma(3,1) - 3*sigma(3,2) + 1*sigma(3,3) = 3*4 - 3*10 + 1*28 = 10;
a(4) = -4*sigma(4,1) + 6*sigma(4,2) - 4*sigma(4,3) + 1*sigma(4,4) = -4*7 + 6*21 - 4*73 + 1*273 = 79.

Crossrefs

Cf. A206763 (exp), A205815, A205812.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * (-1)^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 25 2024 *)
  • PARI
    {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*(-1)^(n-k))}
  • PARI
    {a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-(-x)^k)/(1-(k-1)^k*x^k +x*O(x^n)))), n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Oct 25 2024

A377331 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k,n).

Original entry on oeis.org

1, 7, 58, 707, 11186, 219202, 5097205, 137036819, 4179577045, 142539843882, 5374034016858, 221930535785918, 9962431381720780, 482997720973917947, 25151350530268841003, 1400042027334939211235, 82960609980815501293708, 5213812927633674297808237, 346394632975721545946690108
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 25 2024

Keywords

Crossrefs

Cf. A109974, A205812.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[n, k], {k, 1, n}], {n, 1, 20}]
  • PARI
    {a(n)=sum(k=1, n, binomial(n, k)*sigma(k, n))}

Formula

a(n) ~ n^n / (sqrt(1 + LambertW(exp(-1))) * exp(n) * LambertW(exp(-1))^n).
Showing 1-7 of 7 results.

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