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.

User: Greg Huber

Greg Huber's wiki page.

Greg Huber has authored 58 sequences. Here are the ten most recent ones:

A383472 a(n) = round(Product_{k=1..n} 1 + 2^2^(-k)).

Original entry on oeis.org

1, 2, 5, 11, 23, 46, 92, 184, 369, 738, 1477, 2954, 5909, 11818, 23637, 47274, 94548, 189096, 378193, 756387, 1512775, 3025550, 6051101, 12102203, 24204406, 48408812, 96817625, 193635250, 387270501, 774541002, 1549082004, 3098164009, 6196328018, 12392656037
Offset: 0

Views

Author

Greg Huber, Apr 27 2025

Keywords

Crossrefs

Cf. A215016, A014571.

Programs

  • Mathematica
    Floor[Product[(1+2^(2^(-k))),{k,1,n}]+1/2]
  • PARI
    a(n) = round(prod(k=1, n, 1 + 2^2^(-k))); \\ Michel Marcus, Apr 28 2025

A383471 a(n) = round(Product_{k=1..n} (1 + 1/2^2^(-k))).

Original entry on oeis.org

1, 2, 3, 6, 12, 23, 46, 93, 185, 370, 739, 1478, 2955, 5910, 11819, 23637, 47274, 94549, 189097, 378194, 756388, 1512776, 3025551, 6051102, 12102203, 24204407, 48408813, 96817626, 193635251, 387270501, 774541003, 1549082005, 3098164010, 6196328019, 12392656038, 24785312075
Offset: 0

Views

Author

Greg Huber, Apr 27 2025

Keywords

Crossrefs

Cf. A014571, A215016.

Programs

  • Mathematica
    Floor[Product[(1+2^(-2^(-k))),{k,1,n}]+1/2]
  • PARI
    a(n) = round(prod(k=1, n, 1 + 1/2^2^(-k))); \\ Michel Marcus, Apr 28 2025

A356586 Number of binary digits in the n-th Gosper hyperfactorial of n (A330716).

Original entry on oeis.org

1, 1, 5, 51, 657, 9722, 166296, 3253365, 71905965, 1775175455, 48467529392, 1451065354742, 47289516677131, 1667001471950287, 63213921938077523, 2566296044236261518, 111065406214766719510, 5105032675471072965466, 248377281869637961805657
Offset: 0

Views

Author

Greg Huber, Aug 13 2022

Keywords

Comments

The 0th Gosper hyperfactorial is the usual factorial function.

Examples

			a(0)=1 since 0! has 1 binary digit.
a(3)=51 since the 3rd Gosper hyperfactorial of 3 in binary is 110111011110111100100000111011111111011101100000000, which has 51 digits.

Crossrefs

Cf. A070939, A330716, A356585.

Programs

  • Mathematica
    Floor[Table[1+Sum[Log[k]*(k^n)/Log[2], {k, 1, n}], {n, 1, 18}]]
  • PARI
    a(n) = floor(sum(k=1, n, log(k)*k^n/log(2))) + 1; \\ Michel Marcus, Sep 27 2022

Formula

a(n) = A070939(A330716(n)).

A356585 Number of decimal digits in the n-th Gosper hyperfactorial of n (A330716).

Original entry on oeis.org

1, 1, 2, 16, 198, 2927, 50060, 979361, 21645853, 534381060, 14590180163, 436814197446, 14235563000269, 501817445873045, 19029286646922723, 772532087068933899, 33434018751249535666, 1536767964161539414904, 74769012084248550773909
Offset: 0

Views

Author

Greg Huber, Aug 13 2022

Keywords

Comments

The 0th Gosper hyperfactorial is the usual factorial function.

Examples

			a(0)=1 since the 0th Gosper hyperfactorial (0!) has one decimal digit.
a(3)=16 since the 3rd Gosper hyperfactorial of 3 is 1952152956156672.

Crossrefs

Cf. A055642, A330716, A356586.

Programs

  • Mathematica
    Floor[Table[1+Sum[Log10[k]*(k^n), {k, 1, n}], {n, 1, 18}]]
  • PARI
    a(n) = floor(sum(k=1, n, log(k)*k^n/log(10))) + 1; \\ Michel Marcus, Sep 27 2022

Formula

a(n) = A055642(A330716(n)).

A353321 a(n) = floor(1/erfc(n/sqrt(2))).

Original entry on oeis.org

1, 3, 21, 370, 15787, 1744277, 506797345, 390682215445, 803734397655347, 4430313100526836692, 65618063552490194383194, 2616897361902846669558232537, 281455127862349591601857362987343, 81737217988908649002650313009555641846, 64155724364921456082725604130103414484969173
Offset: 0

Views

Author

Greg Huber, Jul 04 2022

Keywords

Comments

a(n) is the residual (inverted) of the standard normal distribution in two-sided range of n sigma.
The 1/sqrt(2) factor appears in the argument of the erfc function because the standard normal cumulative distribution function has integrand exp(-(1/2)(x/sigma)^2).
The n=5 value, a(5)=1744277, appeared in discussions of the 5-sigma (two-sided probability) discovery of the Higgs boson at CERN (see articles in Links).

Examples

			For n=2, a(2)=21 corresponds roughly to the statement that 95.5% of normally distributed measurements fall into the range of two sigma (plus and minus), since 1/21 = 1-0.955 (approximately). Nearest-integer version (A275366) is always more accurate (e.g., a(2)=22).

Links

Crossrefs

Cf. A275366 (rounded).

Programs

  • Mathematica
    Table[Floor[1/Erfc[n/Sqrt[2]]], {n, 1, 16}]

Formula

Straightforward asymptotics of erfc (e.g., see Abramowitz and Stegun) gives leading order as a(n) ~ sqrt(Pi/2)*n*exp((1/2)*n^2).

A334000 a(n) = (2*n+1)!! * Sum_{k=0..n} k/(2*k+1).

Original entry on oeis.org

0, 1, 11, 122, 1518, 21423, 340869, 6058980, 119218860, 2575293165, 60628447215, 1545696702270, 42437227275450, 1248581232985275, 39197268410049225, 1307969571015966600, 46233376386927067800, 1725823391345415833625, 67845041198360981737875
Offset: 0

Views

Author

Greg Huber, Apr 11 2020

Keywords

Examples

			a(3) = 122 since 0/1 + 1/3 + 2/5 + 3/7 = 122/105 = 122/(7!!).

Crossrefs

Cf. A004041.

Programs

  • Mathematica
    Table[Sum[k/(2*k+1),{k,0,n}],{n,0,18}]*Table[Product[2*j+1,{j,0,n}],{n,0,18}]
FullSimplify[Table[((n+1)/2 - HarmonicNumber[n + 1/2]/4 - Log[2]/2) * (2*n+1)!!, {n, 0, 20}]] (* Vaclav Kotesovec, Apr 14 2020 *)

Formula

a(n) = (2*n+1)!!*(Sum_{k=0..n} k/(2*k+1)).
Recurrence: a(n) = 4*n*a(n-1)-(2*n-1)^2*a(n-2)+(2*n-1)!!.

A334066 a(n) = (2n-1)!!*(Sum_{k=1..n}k/(2*k-1)).

Original entry on oeis.org

1, 5, 34, 298, 3207, 40947, 605076, 10157220, 190915965, 3971997585, 90613969110, 2249113016430, 60338869272675, 1739831420490975, 53656981894391400, 1762410972384203400, 61421841416041392825, 2263752327235180060125, 87970054921758957890250
Offset: 1

Views

Author

Greg Huber, Apr 13 2020

Keywords

Examples

			a(4)=298 since 1/1+2/3+3/5+4/7=298/105=298/(7!!).

Crossrefs

Cf. A004041.

Programs

  • Mathematica
    Table[Sum[k/(2*k-1), {k, 1, n}], {n, 1, 19}]*Table[Product[2*j-1, {j, 1, n}], {n, 1, 19}]
FullSimplify[Table[(n/2 + HarmonicNumber[n - 1/2]/4 + Log[2]/2) * (2*n-1)!!, {n, 1, 20}]] (* Vaclav Kotesovec, Apr 14 2020 *)

Formula

a(n) = (2n-1)!!*(Sum_{k=1..n}k/(2*k-1)).
Recurrence: a(n) = 2*a(n-1) + (2*n-3)^2*a(n-2) + (2*n-1)!!.

A330716 n-th Gosper hyperfactorial of n.

Original entry on oeis.org

1, 1, 16, 1952152956156672
Offset: 0

Views

Author

Greg Huber, Dec 27 2019

Keywords

Comments

Gosper's m-th hyperfactorial of n is the product 1^(1^m)*2^(2^m)*3^(3^m)*...*n^(n^m).
The 0th hyperfactorial is the factorial function.

Examples

			n=3: a(3) = 1^(1^3)*2^(2^3)*3^(3^3) = 2^8 * 3^27.
a(4) has 198 decimal digits and a(5) has 2927 digits.

References

  • R. W. Gosper, "Fac Fun" (ca. 1979).

Links

Crossrefs

Cf. A000142, A002109, A051675, A255321, A255323, A255344 (0th through 5th Gosper hyperfactorials, respectively).

Programs

  • Mathematica
    nmax:=3; Table[Product[i^(i^n),{i,1,n}],{n,0,nmax}] (* Stefano Spezia, Dec 29 2019 *)

A322852 a(n) = floor(e^(e^(n/e))-1).

Original entry on oeis.org

1, 3, 7, 19, 76, 539, 8873, 505531, 173789117, 801854912659, 157566761299009378, 6992370157470308367564294, 780243941119853800883766562690845097, 7117897437316498836240639309909700742341112693648555
Offset: 0

Views

Author

Greg Huber, Dec 28 2018

Keywords

Crossrefs

Cf. A322848. A322851 is a base-2 analog.

Programs

  • Magma
    [Floor(Exp(Exp(n*Exp(-1))) -1): n in [0..14]]; // G. C. Greubel, Feb 14 2019
  • Mathematica
    Array[Floor[E^(E^(#/E)) - 1] &, 14, 0] (* Michael De Vlieger, Jan 04 2019 *)
  • PARI
    default(realprecision, 10000);
A322852(n) = floor(exp(exp(n/exp(1))) -1); \\ Antti Karttunen, Jan 17 2019
  • Sage
    [floor(exp(exp(n*exp(-1))) -1) for n in (0..14)] # G. C. Greubel, Feb 14 2019

A322851 a(n) = floor(2^(2^(n/2))-1).

Original entry on oeis.org

1, 1, 3, 6, 15, 49, 255, 2544, 65535, 6479346, 4294967295, 41981937869755, 18446744073709551615, 1762483107300123635910219390, 340282366920938463463374607431768211455
Offset: 0

Views

Author

Greg Huber, Dec 28 2018

Keywords

Comments

A051179 is a bisection of this sequence.

Crossrefs

Cf. A051179. A322852 is the base-e analog.

Programs

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