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 1-5 of 5 results.

A117684 Row sums of A117683.

Original entry on oeis.org

1, 2, 3, 13, 11, 49, 27, 141, 523, 3081, 923, 5509, 1371, 7617, 24391, 84933, 14795, 110329, 20859, 142101, 499843, 1858209, 241211, 2312077, 8417451, 70482153, 251680159, 935093181, 95916299, 1102272481, 131510523, 1270525629, 4572551611, 17189356473
Offset: 1

Views

Author

Roger L. Bagula, Apr 12 2006

Keywords

Links

Crossrefs

Cf. A117683.

Programs

  • Magma
    A034386:= func< n | n eq 0 select 1 else LCM(PrimesInInterval(1, n)) >;
[(&+[Binomial(n,k)*A034386(k)*A034386(n-k)/A034386(n): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Jul 21 2023
  • Mathematica
    f[n_]:= If[PrimeQ[n], 1, n];
cf[n_]:= cf[n]= If[n==0, 1, f[n]*cf[n-1]]; (* A049614 *)
T[n_, k_]:= T[n, k]= cf[n]/(cf[k]*cf[n-k]);
a[n_]:= a[n]= Sum[T[n,k], {k,n}];
Table[a[n], {n,40}]
  • SageMath
    @CachedFunction
def A034386(n): return product(nth_prime(j) for j in range(1, 1+prime_pi(n)))
def A117684(n): return sum(binomial(n,k)*A034386(k)*A034386(n-k)/A034386(n) for k in range(1,n+1))
[A117684(n) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

Formula

a(n) = Sum_{k=1..n} A117683(n,k).

Extensions

Description simplified, offset corrected by the Assoc. Eds. of the OEIS, Jun 27 2010

A117733 Sum of the n-th primorial and the n-th compositorial number.

Original entry on oeis.org

2, 3, 7, 10, 34, 54, 234, 402, 1938, 17490, 19590, 209670, 237390, 2933070, 43575630, 696759630, 697240110, 12541643310, 12550832490, 250832355690
Offset: 1

Views

Author

Roger L. Bagula, Apr 14 2006

Keywords

Comments

The primorial numbers A034386 define their exponential generating function
A034386(x) = sum_{n>=0} A034386(n)*x^n/n! = sum_{n>=0} x^n/A049614(n).
The compositorial numbers A049614 define their exponential generating function
A049614(x) = sum_{n>=0} A049614(n)*x^n/n! = sum_{n>=0} x^n/A034386(n).
Padding the values with A034386(n=0)=A049614(n=0)=1 at the beginning,
two special values of these are
A049614(x=1) = 4.5892461266379861713581024207350707369274... and
A034386(x=1) = 2.9200509773161347120925629171120194680027...

Crossrefs

Cf. A034386, A117683.

Programs

  • Mathematica
    f[n_] := If[PrimeQ[n] == True, 1, n] cf[0] = 1; cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1] g[n_] := If[PrimeQ[n] == True, n, 1] p[0] = 1; p[n_Integer?Positive] := p[n] = g[n]*p[n - 1] a=Table[cf[n] + p[n], {n, 1, 20}]

Formula

a(n) = A034386(n)+A049614(n).

Extensions

Offset and A-number corrected; comment rewritten - The Assoc Eds of the OEIS, Oct 20 2010

A117734 Absolute difference between the n-th primorial and the n-th compositorial number.

Original entry on oeis.org

0, 1, 5, 2, 26, 6, 186, 18, 1518, 17070, 14970, 205050, 177330, 2873010, 43515570, 696699570, 696219090, 12540622290, 12531433110, 250812956310
Offset: 1

Views

Author

Roger L. Bagula, Apr 14 2006

Keywords

Crossrefs

Cf. A034386, A117683.

Programs

  • Mathematica
    f[n_] := If[PrimeQ[n] == True, 1, n] cf[0] = 1; cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1] g[n_] := If[PrimeQ[n] == True, n, 1] p[0] = 1; p[n_Integer?Positive] := p[n] = g[n]*p[n - 1] Table[Abs[ -cf[n] + p[n]], {n, 1, 20}]

Formula

a(n) = | A034386(n)-A049614(n)| .

Extensions

Offset and A-number corrected; comment rewritten; information duplicating A117733 removed - The Assoc Eds of the OEIS, Oct 20 2010

A117735 a(n) = n! - primorial(n).

Original entry on oeis.org

0, 0, 0, 0, 18, 90, 690, 4830, 40110, 362670, 3628590, 39914490, 478999290, 6226990770, 87178261170, 1307674337970, 20922789857970, 355687427585490, 6402373705217490, 121645100399132310, 2432902008166940310
Offset: 0

Views

Author

Roger L. Bagula, Apr 14 2006

Keywords

Crossrefs

Cf. A034386, A117683.

Programs

  • Mathematica
    f[n_] := If[PrimeQ[n] == True, 1, n] cf[0] = 1; cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1] g[n_] := If[PrimeQ[n] == True, n, 1] p[0] = 1; p[n_Integer?Positive] := p[n] = g[n]*p[n - 1] Table[n! - p[n], {n, 0, 20}]

Formula

a(0)=0. a(n) = n!-A034386(n) = A000142(n)-A034386(n), n>0.

A117736 factorial(n) - A049614(n).

Original entry on oeis.org

0, 0, 1, 5, 20, 116, 696, 5016, 40128, 361152, 3611520, 39899520, 478794240, 6226813440, 87175388160, 1307630822400, 20922093158400, 355686731366400, 6402361164595200, 121645087867699200, 2432901757353984000
Offset: 0

Views

Author

Roger L. Bagula, Apr 14 2006

Keywords

Crossrefs

Cf. A034386, A117683.

Programs

  • Mathematica
    f[n_] := If[PrimeQ[n] == True, 1, n] cf[0] = 1; cf[n_Integer?Positive] := cf[n] = f[n]*cf[n - 1] g[n_] := If[PrimeQ[n] == True, n, 1] p[0] = 1; p[n_Integer?Positive] := p[n] = g[n]*p[n - 1] Table[n! - cf[n], {n, 0, 20}]

Formula

a(n) =n!- A049614(n) = A000142(n)-A049614(n).

Extensions

Edited by the Assoc. Eds. of the OEIS, Jun 27 2010
Showing 1-5 of 5 results.

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