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.

A120266 Numerator of Sum_{k=0..n} n^k/k!.

Original entry on oeis.org

2, 5, 13, 103, 1097, 1223, 47273, 556403, 10661993, 7281587, 62929017101, 7218065, 60718862681977, 595953719897, 13324966405463, 247016301114823, 28505097599389815853, 549689343118061, 320305944459287485595917
Offset: 1

Views

Author

Alexander Adamchuk, Jun 30 2006

Keywords

Comments

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

Examples

			The first few fractions are 2, 5, 13, 103/3, 1097/12, 1223/5, 47273/72, 556403/315, 10661993/2240, ... = A120266/A214401. - _Petros Hadjicostas_, May 12 2020

Links

Crossrefs

Denominators are A214401. Cf. also A063170, A090878, A119029, A120267, A214402.

Programs

  • Mathematica
    Numerator[Table[Sum[n^k/k!, {k,0,n}], {n,1,30}]]

Formula

a(n) = numerator(Sum_{k=0..n} n^k/k!).
a(n) = A063170(n)/A214402(n) = (n!/A214402(n))*Sum_{k=0..n} n^k/k! for n > 0. - Jonathan Sondow, Jul 16 2012

A119029 Numerator of Sum_{k=1..n} n^(k-1)/k!.

Original entry on oeis.org

1, 2, 4, 25, 217, 203, 6743, 69511, 1184417, 728102, 5720654791, 601499, 4670663321629, 42568060798, 888330615353, 15438515749903, 1676770323947695709, 30538296012677, 16858207434636875406943
Offset: 1

Views

Author

Alexander Adamchuk, Jul 22 2006

Keywords

Comments

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

Examples

			The first few fractions are 1, 2, 4, 25/3, 217/12, 203/5, 6743/72, 69511/315, 1184417/2240, 728102/567, ... = A119029/A214401. - _Petros Hadjicostas_, May 12 2020

Crossrefs

Cf. A063170, A090878, A093101, A120266, A120267, A214401 (denominators), A214402.

Programs

  • Mathematica
    Numerator[Table[Sum[n^(k-1)/k!,{k,1,n}],{n,1,30}]]

Formula

a(n) = numerator(Sum_{k=1..n} n^(k-1)/k!).
a(n) = A120267(n)/n.

A214401 Denominator of Sum_{k=0..n} n^k/k!.

Original entry on oeis.org

1, 1, 1, 3, 12, 5, 72, 315, 2240, 567, 1814400, 77, 239500800, 868725, 7175168, 49116375, 2092278988800, 14889875, 3201186852864000, 14849255421, 3783802880000, 3543572316375, 562000363888803840000, 2505147019375, 496358721386591551488
Offset: 1

Views

Author

Jonathan Sondow, Jul 15 2012

Keywords

Comments

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in the current sequence. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

Links

Crossrefs

Numerators are A120266.
Cf. also A063170, A090878, A093101, A119029, A120267, A214402.

Programs

  • Mathematica
    Denominator[Table[Sum[n^k/k!, {k, 0, n}], {n, 1, 30}]]
  • PARI
    a(n) = denominator(sum(k=0, n, n^k/k!)); \\ Michel Marcus, Apr 20 2021

Formula

a(n) = n!/A214402(n).

A129977 Numbers m such that A119029(m) = numerator(Sum_{k=1..m} m^(k-1)/k!) is prime.

Original entry on oeis.org

2, 17, 102, 112, 316, 447, 535, 820, 1396, 1475, 1650, 5575, 6486, 6832
Offset: 1

Views

Author

Alexander Adamchuk, Jun 13 2007

Keywords

Comments

For n >= 1, the corresponding primes are A119029(a(n)) = {2, 1676770323947695709, ...}.
a(15) > 10000. - Lucas A. Brown, Apr 01 2021

Links

Crossrefs

Cf. A119029, A120266, A120267.

Programs

  • Mathematica
    Do[ f=Numerator[ Sum[ n^(k-1)/k!, {k, 1, n} ] ]; If[ PrimeQ[f], Print[{n,f}] ], {n,1,316} ]
Select[Range[2000],PrimeQ[Numerator[Sum[#^(k-1)/k!,{k,#}]]]&] (* Harvey P. Dale, Jun 15 2019 *)
  • PARI
    for( n=1,1000, if( ispseudoprime( numerator( sum( k=1,n,n^(k-1)/k!))), print1(n", "))) \\ M. F. Hasler, Jun 18 2007

Extensions

Edited and extended (a(6)..a(8)) by M. F. Hasler, Jun 18 2007
More terms from Ryan Propper, Jan 12 2008
Various sections edited by Petros Hadjicostas, May 12 2020
a(12)-a(14) from Lucas A. Brown, Apr 01 2021

A129976 Numbers k such that the numerator of Sum_{j=0..k} k^j/j! is a prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 14, 21, 33, 36, 56, 68, 94, 378, 1943, 2389, 2710, 5455, 6804
Offset: 1

Views

Author

Alexander Adamchuk, Jun 13 2007

Keywords

Comments

The corresponding primes are A120266(a(n)) = {2, 5, 13, 103, 1097, 1223, ...}
a(22) > 10^4. - Michael S. Branicky, Apr 27 2025

Examples

			Sum_{j=0..4} 4^j/j! = 103/3. The numerator is a prime, hence 4 is in the sequence.

Crossrefs

Cf. A120266, A119029, A120267.

Programs

  • Mathematica
    Do[ f=Numerator[Sum[n^k/k!,{k,0,n}]]; If[PrimeQ[f], Print[{n,f}]], {n, 1, 378}]

Extensions

Edited by Stefan Steinerberger, Jul 22 2007
a(17)-a(18) and a(20) from Ryan Propper, Jan 22 2008
a(19) and a(21) from Michael S. Branicky, Apr 26 2025
Showing 1-5 of 5 results.

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