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.

Previous Showing 11-20 of 20 results.

A033149 Position of first occurrence of n in the continued fraction for the Euler-Mascheroni constant (gamma).

Original entry on oeis.org

2, 4, 9, 8, 11, 69, 24, 14, 139, 52, 22, 161, 10, 199, 337, 79, 163, 176, 384, 614, 183, 651, 137, 480, 250, 862, 554, 618, 287, 300, 1952, 166, 150, 2038, 560, 483, 1284, 681, 306, 20, 349, 1130, 2280, 1884, 1903, 2564, 4753, 717, 31, 2610, 568, 248, 2171
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

The smallest positive integers not appearing in the first 970,258,158 terms of the c.f. are 13161, 13295, 14734, 14970, 14971, 15795, 15985, 16011, 16110, ... - Eric W. Weisstein, Sep 21 2011

Links

Crossrefs

Cf. A001620, A002852, A032523.
Cf. A224847 (= a(n) -1).

Programs

  • Mathematica
    With[{cf=ContinuedFraction[EulerGamma,5000]},Table[Position[cf,n,1,1],{n,60}]]//Flatten (* Harvey P. Dale, Aug 06 2025 *)
  • PARI
    /* 15000 precision digits */ v=contfrac(Euler); a(n)=if(n<0,0,s=1; while(abs(n-component(v,s))>0,s++); s)

Formula

a(n) = A224847(n) + 1.

Extensions

More terms from Benoit Cloitre, Oct 20 2002

A059856 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into minimal chunks so that chunks have increasing length and do not begin with zero.

Original entry on oeis.org

5, 77, 215, 66490, 1532860, 6065120900, 82402431042, 159335939923, 5988057672348, 84867726777664, 670936947063291, 7467495146314472, 498070824809605040, 1448654283622417399, 76449235362535003337
Offset: 0

Views

Author

Jason Earls, Feb 27 2001

Keywords

Examples

			0.5772156649015328606065120900824024310421593359399235...

Crossrefs

Cf. A001620, A002852, A033091, A098967.

Extensions

More terms from Tracy Poff (tracy.poff(AT)gmail.com), Apr 15 2005

A066034 Second term in the continued fraction expansion of StieltjesGamma[n].

Original entry on oeis.org

1, -13, -103, 486, 430, 1260, -4188, -1896, -2839, -29074, 4870, 3701, 5978, -36411, -4779, -3527, -5007, 38056, 3253, 1985, 2144, 9575, -1846, -803, -629, -930, 1522, 287, 156, 135, 281, -133, -38, -22, -19, -49, 13, 4, 2, 1, 4, -1, -1, -3, -5, -13, 1, 25, 1, 1, 1, -5, -5, -1, -2, -7, -3, 3, 2, 3, 3, 1, 1, -1
Offset: 0

Views

Author

Wouter Meeussen, Dec 12 2001

Keywords

Comments

StieltjesGamma[0] equals EulerGamma

Examples

			The continued fraction expansions of the first four unsigned StieltjesGamma[n], n=0..3, are {0,1,1,2,1,2,1,4,3,13,5,1}, {0,13,1,2,1,2,1,74,1,10,1,9}, {0,103,5,8,3,9,1,8,10,1,10,1},{0,486,1,8,2,4,2,1,1,3,1,2}

Crossrefs

Cf. A002852, A066035, A066036.

Programs

  • Mathematica
    Part[ #, 2 ]&/@Table[ ContinuedFraction[ StieltjesGamma[ n ]~N~24, 12 ], {n, 0, 64} ]

A066035 First term in the continued fraction expansion of StieltjesGamma[n].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -5, -7, -5, 6, 34, 78, 125, 126, -19, -463, -1340, -2572, -3457, -2055, 5372, 24019, 57424, 98543, 111670, 5333, -390972, -1303180, -2845076, -4540526, -4341905, 2871566, 26604908, 79321663
Offset: 0

Views

Author

Wouter Meeussen, Dec 12 2001

Keywords

Comments

StieltjesGamma[0] equals EulerGamma

Examples

			The continued fractions of the StieltjesGamma[n], n=42..45, are {-2,-1,-1,-1,-3,-3,-7},{-5,-3,-1,-3,-2,-3,-2}, {-7,-5,-3,-2,-1,-4,-1},{-5,-13,-1,-4,-1,-1,-1}

Crossrefs

Cf. A002852, A066034, A066036.

Programs

  • Mathematica
    Part[ #, 1 ]&/@Table[ ContinuedFraction[ StieltjesGamma[ n ]~N~24, 12 ], {n, 0, 81} ]

A066036 Continued fraction expansion of StieltjesGamma[1].

Original entry on oeis.org

0, 13, 1, 2, 1, 2, 1, 74, 1, 10, 1, 9, 2, 1, 3, 1, 4, 1, 6, 1, 1, 2, 84, 1, 108, 1, 20, 22, 2, 2, 1, 2, 2, 1, 7, 1, 66, 2, 1, 1, 2, 5, 1, 1, 2, 1, 1, 59, 1, 2, 1, 5, 19, 3, 3, 1, 5, 4, 4, 1, 1, 4, 2, 32
Offset: 0

Views

Author

Wouter Meeussen, Dec 12 2001

Keywords

Examples

			StieltjesGamma[1] = -0.07281584548367672486058637587490131913773633833433...

Crossrefs

Cf. A002852, A066034, A066035.

Programs

  • Mathematica
    ContinuedFraction[Abs@StieltjesGamma[n]~N~100, 64]
  • PARI
    contfrac(intnum(x=0, oo, (1/tanh(Pi*x)-1)*(x*log(1+x^2)-2*atan(x))/(2*(1+x^2)))) \\ Charles R Greathouse IV, Mar 10 2016

A224847 Position of first occurrence of n in the continued fraction for the Euler-Mascheroni constant (gamma).

Original entry on oeis.org

1, 3, 8, 7, 10, 68, 23, 13, 138, 51, 21, 160, 9, 198, 336, 78, 162, 175, 383, 613, 182, 650, 136, 479, 249, 861, 553, 617, 286, 299, 1951, 165, 149, 2037, 559, 482, 1283, 680, 305, 19, 348, 1129, 2279, 1883, 1902, 2563, 4752, 716, 30, 2609, 567, 247, 2170, 7776
Offset: 1

Views

Author

Eric W. Weisstein, Jul 22 2013

Keywords

Comments

This sequence is the same as A033149, but uses correct [a_0; a_1, a_2, ...] indexing of continued fraction terms.
The smallest numbers not occurring in the first 4,851,382,841 terms of the c.f. are 27943, 33436, 33978, 34017, ... - Eric W. Weisstein, Jul 22 2013

Examples

			The c.f. for gamma is  A002852 = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, ...].
1 occurs first at term a_1
2 occurs first at term a_3.
3 occurs first at term a_8.
4 occurs first at term a_7.

Links

Crossrefs

Cf. A033149(n) = a(n) + 1.
Cf. A002852 (continued fraction for Euler-Mascheroni constant).

Programs

  • Mathematica
    With[{cfeg=Rest[ContinuedFraction[EulerGamma,8000]]},Table[Position[cfeg,n,1,1],{n,60}]]//Flatten (* Harvey P. Dale, Nov 06 2024 *)

A080410 Boustrophedon transform of the continued fraction of the Euler-Mascheroni constant, gamma (A001620).

Original entry on oeis.org

0, 1, 3, 8, 23, 72, 279, 1236, 6313, 36133, 230119, 1611138, 12308693, 101865629, 907900133, 8669791288, 88309821406, 955736037556, 10951928988000, 132472073263683, 1686686835102650, 22549341913109430, 315817852408881670
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 18 2003

Keywords

Examples

			We simply apply the Boustrophedon transform to [0,1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,...] (A002852)

Links

Crossrefs

Cf. A001620, A002852, A080406-A080409.

Formula

a(n) appears to be asymptotic to C*n!*(2/Pi)^n where C=5.79838940503783299259552225238077705314049166104773668246015... which almost satisfies the polynomial equation 94487-16249C-8C^2=0 - Benoit Cloitre and Mark Hudson (mrmarkhudson(AT)hotmail.com)

A184977 a(n) = Sum_{k=1..n} floor(k*gamma) where gamma is Euler's constant (A001620).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 13, 17, 22, 27, 33, 39, 46, 54, 62, 71, 80, 90, 100, 111, 123, 135, 148, 161, 175, 190, 205, 221, 237, 254, 271, 289, 308, 327, 347, 367, 388, 409, 431, 454, 477, 501, 525, 550, 575, 601, 628, 655, 683, 711, 740, 770, 800, 831, 862, 894, 926, 959, 993, 1027, 1062
Offset: 1

Views

Author

Michel Lagneau, Mar 27 2011

Keywords

Comments

a(n) = A183143(n) for n = 1..96 where A183143(n) is the sequence floor(1/r) + floor(2/r) + ... + floor(n/r) and r = sqrt(3). It is interesting to note that a(n)/n^2 converges to gamma/2.
gamma = 0.57721566490153286060651209... (A002852)
1/sqrt(3) = 0.577350269189625764509148... (A020760)
Starts to differ from A183143 at a(97). - R. J. Mathar, Aug 28 2025

Links

Crossrefs

Cf. A001620, A038128.

Programs

  • Magma
    R:=RealField(100); [(&+[Floor(k*EulerGamma(R)): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Aug 27 2018
  • Maple
    with(numtheory):Digits:=500:s:=0:c:=evalf(gamma(0)):for n from 1 to 100 do:
  s:=s+floor(n*c):printf(`%d, `,s):od:
  • Mathematica
    Table[Sum[Floor[k*EulerGamma], {k, 1, n}], {n, 50}] (* G. C. Greubel, Jun 02 2017 *)
  • PARI
    a(n) = sum(k=1, n, floor(k*Euler)); \\ Michel Marcus, Apr 02 2017

Formula

Partial sums of A038128.

Extensions

Name edited by Jon E. Schoenfield, Apr 02 2017

A178783 Continued fraction for Euler-Mascheroni constant with convergents 0/1, 1/1, 1/2, 4/7, etc., which lie between the monotonically increasing series given by (Sum_{k=1..n} 1/k - Sum_{k=n..n^2} 1/k) and the monotonically decreasing series (Sum_{k=1..n} 1/k - Sum_{k=n..n^2-1} 1/k), both of which converge to gamma. Thus each p/q in the sequence lies within 1/q^2 of gamma.

Original entry on oeis.org

0, 1, 1, 3, -4, -5, 3, 13, 5, 2, -10, -3, 4, 2, -42, -12, 3, 8, -9, -2, 6, -50, 5, -67, -5, 7, 12, -401, -2, -2, 3, 3, -4, -6, 3, 3, -12, -3, -2, 2, 2, -5, -6
Offset: 0

Views

Author

Joseph G. Johnson (jjohnson1253(AT)hotmail.com), Jun 12 2010

Keywords

Comments

Series derived from def. gamma = lim(Sum_{k=1..n} 1/k - log(n)) by noting that 2*gamma = 2*Sum_{k=1..n} 1/k - 2*log(n) (ignoring limit) and also gamma = Sum_{k=1..n^2} 1/k - log(n^2), then gamma = 2*gamma - gamma gets rid of the log term and the series consists of all rational terms. The decreasing series was found by accident. The proofs for both are straightforward. The PARI program uses the first term of the Euler-Maclaurin summation and gamma itself for the upper and lower bounds.

Crossrefs

Cf. A002852.

Programs

  • PARI
    pconv=vector(43); qconv=vector(43); cf=vector(43); fract=vector(43); pconv[1]=0; pconv[2]=1; pconv[3]=1; pconv[4]=4; qconv[1]=1; qconv[2]=1; qconv[3]=2; qconv[4]=7; cf[1]=0; cf[2]=1; cf[3]=1; cf[4]=3; fract[1]=0/1; fract[2]=1/1; fract[3]=1/2; fract[4]=4/7; for(k=5,43, tst=0; cfm=1; until(tst==1, pp = cfm * pconv[k - 1] + pconv[k - 2]; pn = cfm * pconv[k - 1] - pconv[k - 2]; qp = cfm * qconv[k - 1] + qconv[k - 2]; qn = cfm * qconv[k - 1] - qconv[k - 2]; slp = pp/qp; sln = pn/qn; if(((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) ||
(slp - Euler < 1/(3 * qp^2) && slp - Euler > 0)) || ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0)), pconv[k] = ((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) || (slp - Euler < 1/(3 * qp^2) && slp - Euler > 0))*pp + ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0))*pn; qconv[k] = ((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) ||
(slp - Euler < 1/(3 * qp^2) && slp - Euler > 0))*qp + ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0))*qn; fract[k] = pconv[k]/qconv[k]; cf[k] = ((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) || (slp - Euler < 1/(3 * qp^2) && slp - Euler > 0))*cfm - ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0))*cfm; tst = 1, cfm = cfm + 1)); write("eulwritefile.txt","Convergents: ",fract); write("eulwritefile.txt","continued fraction: ",cf); write("eulwritefile.txt","sln: ",sln); write("eulwritefile.txt","slp: ",slp))

A346525 Decimal expansion of gamma/(1 - gamma), where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

1, 3, 6, 5, 2, 7, 2, 1, 1, 8, 6, 2, 5, 4, 4, 1, 5, 5, 1, 8, 7, 7, 2, 1, 9, 3, 2, 8, 4, 5, 6, 5, 2, 2, 0, 1, 6, 1, 8, 8, 3, 1, 6, 0, 7, 1, 7, 6, 5, 1, 1, 1, 0, 5, 8, 6, 9, 9, 7, 6, 7, 6, 9, 4, 9, 5, 1, 0, 3, 4, 3, 7, 5, 4, 2, 1, 0, 2, 8, 8, 0, 0, 0, 7, 0
Offset: 1

Views

Author

Christoph B. Kassir, Jul 26 2021

Keywords

Comments

Apart from the first digit the same as A091556. - R. J. Mathar, Aug 23 2021

Examples

			1.36527211862544155187721932845652201618831607176511...

Crossrefs

Cf. A001620, A346589.
Cf. A002852 (continued fraction of gamma; omit first two terms for continued fraction of gamma/(1 - gamma)).

Programs

  • Mathematica
    RealDigits[EulerGamma/(1 - EulerGamma), 10, 100][[1]] (* Alonso del Arte, Jul 26 2021 *)
  • PARI
    Euler/(1-Euler)
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