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 1-10 of 29 results. Next

A051229 Numbers m such that the Bernoulli number B_{2*m} has denominator 66.

Original entry on oeis.org

5, 25, 85, 185, 235, 295, 305, 335, 355, 365, 395, 425, 505, 535, 635, 685, 695, 745, 815, 835, 925, 985, 995, 1115, 1135, 1145, 1285, 1315, 1345, 1385, 1415, 1445, 1475, 1525, 1535, 1555, 1565, 1585, 1655, 1675, 1735, 1765
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

Examples

			The numbers m = 5, 25 belong to the list because B_10 = 5/66 and B_50 = 495057205241079648212477525/66. - _Petros Hadjicostas_, Jun 06 2020

References

  • B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228.
Equals A051230/2.

Programs

  • Mathematica
    Select[Range[2000],Denominator[BernoulliB[2 #]]==66&] (* Harvey P. Dale, Mar 11 2012 *)
  • PARI
    is(n)=denominator(bernfrac(2*n))==66 \\ Charles R Greathouse IV, Feb 06 2017
  • Sage
    [n for n in (1..2000) if denominator(bernoulli(2*n))==66 ] # G. C. Greubel, Jun 06 2020

Formula

a(n) = 5*A119456(n). - G. C. Greubel, Jun 06 2020

Extensions

More terms from Michael Somos
Name edited by Petros Hadjicostas, Jun 06 2020

A271634 Numbers n such that Bernoulli number B_{n} has denominator 510.

Original entry on oeis.org

16, 32, 64, 128, 304, 496, 608, 752, 944, 992, 1504, 1648, 1744, 1984, 2512, 2672, 3008, 3152, 3296, 3376, 3488, 3568, 3632, 3664, 3856, 3968, 4112, 4208, 4528, 4976, 5024, 5072, 5344, 5584, 5648, 5776, 5872, 6016, 6064, 6128, 6224, 6304, 6592, 6752, 7024, 7136, 7264
Offset: 1

Views

Author

Paolo P. Lava, Apr 21 2016

Keywords

Comments

510 = 2 * 3 * 5 * 17.
All terms are multiple of a(1) = 16.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 463.

Examples

			Bernoulli B_{16} is -3617/510, hence 16 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,510);
  • Mathematica
    Select[Range[0, 1000], Denominator[BernoulliB[#]] == 510 &] (* Robert Price, Apr 21 2016 *)
  • PARI
    isok(n) = denominator(bernfrac(n)) == 510; \\ Michel Marcus, Apr 22 2016

Extensions

More terms from Michel Marcus, Apr 22 2016

A271635 Numbers n such that Bernoulli number B_{n} has denominator 138.

Original entry on oeis.org

22, 154, 242, 286, 374, 814, 1034, 1078, 1298, 1342, 1474, 1562, 1694, 1738, 2134, 2222, 2354, 2794, 3014, 3058, 3146, 3278, 3454, 3586, 3674, 3982, 4114, 4246, 4334, 4378, 4906, 4994, 5654, 5698, 5786, 5918, 5962, 6094, 6226, 6754, 6842, 6886, 6974, 7414, 7634, 7678, 7766
Offset: 1

Views

Author

Paolo P. Lava, Apr 21 2016

Keywords

Comments

138 = 2 * 3 * 23.
All terms are multiple of a(1) = 22.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 17.

Examples

			Bernoulli B_{22} is 854513/138, hence 22 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,138);
  • Mathematica
    Select[Range[0, 1000], Denominator[BernoulliB[#]] == 138 &] (* Robert Price, Apr 21 2016 *)
  • PARI
    isok(n) = denominator(bernfrac(n)) == 138; \\ Michel Marcus, Apr 22 2016

Extensions

More terms from Michel Marcus, Apr 22 2016

A272138 Numbers n such that Bernoulli number B_{n} has denominator 798.

Original entry on oeis.org

18, 54, 342, 558, 774, 1026, 1206, 1674, 1962, 2322, 2826, 2934, 3006, 3474, 3618, 3798, 4014, 4086, 4122, 4842, 5706, 5886, 6282, 6354, 6498, 6894, 7002, 7362, 7578, 7794, 7902, 8082, 8226, 8334, 8478, 8766, 8982, 9018, 9378, 9414, 9846, 10134, 10278, 10422, 10602, 10782
Offset: 1

Views

Author

Paolo P. Lava, Apr 21 2016

Keywords

Comments

798 = 2 * 3 * 7 * 19.
All terms are multiple of a(1) = 18.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 775.

Examples

			Bernoulli B_{18} is 43867/798, hence 18 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272139, A272140, A272183, A272184, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,798);
  • Mathematica
    Select[Range[0, 1000], Denominator[BernoulliB[#]] == 798 &] (* Robert Price, Apr 21 2016 *)
  • PARI
    lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 798, print1(n, ", "))); \\ Altug Alkan, Apr 22 2016

Extensions

More terms from Altug Alkan, Apr 22 2016

A272139 Numbers n such that Bernoulli number B_{n} has denominator 1806.

Original entry on oeis.org

42, 294, 798, 1806, 2058, 2814, 2982, 4074, 4578, 5334, 5586, 6594, 6846, 8106, 8274, 8358, 9366, 9534, 12642, 12894, 13314, 14154, 14658, 15162, 17178, 18186, 19194, 20118, 20454, 21882, 21966, 22722, 22974, 23982, 25914, 26502, 27006, 28266, 28518, 29778
Offset: 1

Views

Author

Paolo P. Lava, Apr 21 2016

Keywords

Comments

1806 = 2 * 3 * 7 * 43.
All terms are multiple of a(1) = 42.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 1.
In 2005, B. C. Kellner proved E. W. Weisstein's conjecture that denom(B_n) = n only if n = 1806.

Examples

			Bernoulli B_{42} is 1520097643918070802691/1806, hence 42 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272140, A272183, A272184, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,1806);
  • Mathematica
    Select[Range[0, 1000], Denominator[BernoulliB[#]] == 1806 &] (* Robert Price, Apr 21 2016 *)
Select[Range[42,30000,42],Denominator[BernoulliB[#]]==1806&] (* Harvey P. Dale, Jun 01 2019 *)
  • PARI
    lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 1806, print1(n, ", "))); \\ Altug Alkan, Apr 22 2016

Extensions

More terms from Altug Alkan, Apr 22 2016

A272140 Numbers n such that Bernoulli number B_{n} has denominator 1590.

Original entry on oeis.org

52, 104, 988, 1976, 3068, 3172, 5252, 5356, 5564, 6136, 6344, 7124, 7748, 8164, 8684, 10244, 10712, 12532, 13364, 13676, 13988, 14092, 16276, 16328, 17212, 17368, 17524, 18044, 18356, 19084, 19916, 20228, 20488, 20644, 22828, 23348, 23764
Offset: 1

Views

Author

Paolo P. Lava, Apr 21 2016

Keywords

Comments

1590 = 2 * 3 * 5 * 53.
All terms are multiple of a(1) = 52.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 1507.

Examples

			Bernoulli B_{52} is -801165718135489957347924991853/1590, hence 52 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272183, A272184, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,1590);
  • Mathematica
    Select[Range[0, 1000], Denominator[BernoulliB[#]] == 1590 &] (* Robert Price, Apr 21 2016 *)
  • PARI
    isok(n) = denominator(bernfrac(n)) == 1590; \\ Michel Marcus, Apr 22 2016

Extensions

a(12)-a(15) from Michel Marcus, Apr 22 2016
More terms from Altug Alkan, Apr 22 2016

A272183 Numbers n such that Bernoulli number B_{n} has denominator 330.

Original entry on oeis.org

20, 340, 1220, 1420, 2020, 2980, 3340, 3940, 4460, 4540, 4580, 5140, 5660, 5780, 6260, 6340, 6620, 6940, 7060, 7580, 7660, 7780, 7940, 8020, 8980, 9140, 9260, 9580, 10420, 10820, 11140, 11380, 11740, 12140, 12340, 12860, 13220, 13540, 14020, 15020, 15140, 15740, 15940, 16540, 16780
Offset: 1

Views

Author

Paolo P. Lava, Apr 22 2016

Keywords

Comments

330 = 2 * 3 * 5 * 11.
All terms are multiple of a(1) = 20.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 289.

Examples

			Bernoulli B_{20} is -174611/330, hence 20 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272184, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,330);
  • Mathematica
    Select[20 Range@ 850, Denominator@ BernoulliB@ # == 330 &] (* Michael De Vlieger, Apr 29 2016 *)
  • PARI
    isok(n) = denominator(bernfrac(n)) == 330; \\ Michel Marcus, Apr 22 2016

Extensions

a(15)-a(29) from Michel Marcus, Apr 22 2016
a(30)-a(45) from Altug Alkan, Apr 22 2016

A272184 Numbers n such that Bernoulli number B_{n} has denominator 282.

Original entry on oeis.org

46, 322, 782, 874, 1058, 1702, 2162, 2254, 2714, 2806, 3266, 3634, 4646, 4738, 4922, 5014, 6118, 6302, 6394, 6854, 7222, 7406, 7682, 8326, 8878, 9062, 9154, 9706, 10442, 10534, 11822, 11914, 12098, 12374, 12466, 13018, 13294, 14122, 14306, 14398, 14582
Offset: 1

Views

Author

Paolo P. Lava, Apr 22 2016

Keywords

Comments

282 = 2 * 3 * 47.
All terms are multiples of a(1) = 46.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 41.

Examples

			Bernoulli B_{46} is 596451111593912163277961/282, hence 46 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272185, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,282);
  • Mathematica
    Select[46 Range@ 320, Denominator@ BernoulliB@ # == 282 &] (* Michael De Vlieger, Apr 29 2016 *)
  • PARI
    isok(n) = denominator(bernfrac(n)) == 282; \\ Michel Marcus, Apr 22 2016

Extensions

a(13)-a(28) from Michel Marcus, Apr 22 2016
a(29)-a(41) from Altug Alkan, Apr 22 2016

A272185 Numbers n such that Bernoulli number B_{n} has denominator 870.

Original entry on oeis.org

28, 56, 532, 868, 1064, 1736, 1988, 2828, 2884, 3052, 3836, 5068, 5516, 5768, 5908, 6104, 6244, 6356, 6412, 6748, 7196, 7364, 7924, 8708, 8764, 8876, 9268, 9716, 9772, 10108, 10136, 10276, 10724, 10892, 11032, 11228, 11816, 12292, 12488, 12796, 12824, 12908, 12964, 13076, 13412, 13496, 14392
Offset: 1

Views

Author

Paolo P. Lava, Apr 22 2016

Keywords

Comments

870 = 2 * 3 * 5 * 29.
All terms are multiple of a(1) = 28.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 811.

Examples

			Bernoulli B_{28} is -23749461029/870, hence 28 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272186.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,870);
  • Mathematica
    Select[28 Range@ 520, Denominator@ BernoulliB@ # == 870 &] (* Michael De Vlieger, Apr 29 2016 *)
  • PARI
    isok(n) = denominator(bernfrac(n)) == 870; \\ Michel Marcus, Apr 22 2016

Extensions

a(13)-a(29) from Michel Marcus, Apr 22 2016
More terms from Altug Alkan, Apr 22 2016

A272186 Numbers n such that Bernoulli number B_{n} has denominator 690.

Original entry on oeis.org

44, 484, 748, 2596, 2684, 3124, 4444, 4708, 6556, 6908, 7964, 8228, 9812, 9988, 11308, 11572, 11836, 11924, 12452, 13684, 13772, 13948, 14828, 15356, 15532, 16148, 16676, 16852, 17468, 17644, 18524, 19316, 19756, 20108, 20284, 20372, 21076, 22924, 23012, 24068, 24772, 25124, 25828, 26444
Offset: 1

Views

Author

Paolo P. Lava, Apr 22 2016

Keywords

Comments

690 = 2 * 3 * 5 * 23.
All terms are multiple of a(1) = 44.
For these numbers Numerator(B_{n}) mod Denominator(B_{n}) = 637.

Examples

			Bernoulli B_{44} is -27833269579301024235023/690, hence 44 is in the sequence.

Links

Crossrefs

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local n;  for n from 2 by 2 to q do
if denom(bernoulli(n))=h then print(n); fi; od; end: P(10^6,690);
  • PARI
    isok(n) = denominator(bernfrac(n)) == 690; \\ Michel Marcus, Apr 22 2016

Extensions

a(9)-a(14) from Michel Marcus, Apr 22 2016
More terms from Altug Alkan, Apr 22 2016
Showing 1-10 of 29 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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