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.

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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

A272369 Numbers n such that Bernoulli number B_{n} has denominator 1410.

Original entry on oeis.org

92, 184, 1564, 1748, 2116, 3496, 4232, 4324, 5428, 5612, 6532, 8648, 9476, 9844, 10028, 10856, 11224, 12604, 14444, 15364, 16652, 18124, 18952, 19412, 20056, 20884, 21068, 23644, 24932, 26036, 26588, 28612, 28796, 28888, 29164, 30728, 31004, 31924, 32108, 32476, 33304, 34868, 35236, 35788, 36248, 36524
Offset: 1

Views

Author

Paolo P. Lava, Apr 28 2016

Keywords

Comments

1410 = 2 * 3 * 5 * 47.
All terms are multiple of a(1) = 92.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 1333.

Examples

			Bernoulli B_{92} is -1295585948207537527989427828538576749659341483719435143023316326829946247/1410, hence 92 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, 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(10^6,1410);
  • Mathematica
    Select[92 Range@ 360, Denominator@ BernoulliB@ # == 1410 &] (* Michael De Vlieger, Apr 28 2016 *)
  • PARI
    lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 1410, print1(n, ", "))); \\ Altug Alkan, Apr 28 2016
  • Python
    from sympy import divisors, isprime
A272369_list = []
for i in range(92, 10**6, 92):
    for d in divisors(i):
        if d not in (1,2,4,46) and isprime(d+1):
            break
    else:
        A272369_list.append(i) # Chai Wah Wu, May 02 2016

Extensions

a(18)-a(29) from Altug Alkan, Apr 28 2016
More terms from Michael De Vlieger, Apr 28 2016

A282773 Numbers n such that Bernoulli number B_{n} has denominator 498.

Original entry on oeis.org

82, 574, 1066, 1394, 3034, 3362, 3854, 4838, 5494, 5822, 6478, 7462, 7954, 8282, 8774, 8938, 10414, 11234, 12218, 12382, 12874, 13694, 15826, 16154, 17302, 18614, 18778, 21074, 21238, 21566, 22058, 22222, 22714, 23206, 23534, 23698, 25174, 25502, 25994
Offset: 1

Views

Author

Paolo P. Lava, Mar 07 2017

Keywords

Comments

498 = 2 * 3 * 83.
All terms are multiples of a(1) = 82.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 77.
n such that 82 | n but there are no primes p other than 2, 3, 83 such that p-1 | n. - Robert Israel, Mar 07 2017

Examples

			Bernoulli B_{82} is 1677014149185145836823154509786269900207736027570253414881613/498, hence 82 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, A272186, A272369.
Cf. A002445.

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(10^6,498);
# Alternative:
filter:= n ->
  select(isprime,map(`+`,numtheory:-divisors(n),1)) = {2,3,83}:
select(filter, [seq(i,i=82..10^5,82)]); # Robert Israel, Mar 07 2017
  • Mathematica
    Select[82 Range[360], Denominator@ BernoulliB@ # == 498 &] (* Michael De Vlieger, Mar 07 2017 *)

Extensions

More terms from Michael De Vlieger, Mar 07 2017

A169980 Numerator(Bernoulli(2n)) mod denominator(Bernoulli(2n)).

Original entry on oeis.org

0, 1, 29, 1, 29, 5, 2039, 1, 463, 775, 289, 17, 2039, 1, 811, 12899, 463, 1, 1280537, 1, 11519, 1, 637, 41, 31933, 5, 1507, 775, 811, 53, 34488049, 1, 463, 62483, 29, 289, 91560011, 1, 29, 37, 182293, 77, 2346073, 1, 56003, 230759, 1333, 1, 3051091, 1, 28859, 61, 1507
Offset: 0

Views

Author

Robert G. Wilson v, Aug 19 2010

Keywords

Comments

From Robert G. Wilson v, Aug 27 2010: (Start)
From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.
Values sorted: 1, 5, 17, 29, 37, 41, 49, 53, 61, 65, 77, 101, 137, 161, 169, 173, 181, 185, 221, 229, ..., .
a(n)== 1 for n's: 1, 3, 7, 13, 17, 19, 21, 31, 37, 43, 47, 49, 57, 59, 61, 67, 71, 73, 79, 91, 93, 97, ..., .
a(n)== 5 for n's: 5, 25, 85, 185, 235, 295, 305, 335, 355, 365, 395, 425, 505, 535, 635, 685, 695, ..., . A051229
a(n)==17 for n's: 11, 77, 87, 121, 143, 187, 407, 517, 539, 649, 671, 737, 781, 847, 869, 1067, 1111, ..., .
a(n)==29 for n's: 2, 4, 34, 38, 62, 76, 94, 118, 122, 124, 142, 188, 202, 206, 214, 218, 236, 244, ..., . A051225
a(n)==37 for n's: 39, 507, 1209, 1677, 3783, 4251, 5421, 5811, 6123, 6357, 6513, 7526, 7682, 7760, 8228, ..., .
a(n)==41 for n's: 23, 123, 161, 391, 437, 529, 851, 1081, 1127, 1357, 1403, 1633, 1817, 2323, 2369, 2461, ..., .
a(n)==49 for n's: 55, 275, 605, 2035, 3025, 3355, 3685, 3905, 4345, 5555, 5885, 6985, 7535, 7645, 8195, ..., .
a(n)==53 for n's: 29, 203, 377, 493, 841, 899, 1073, 1247, 1363, 1711, 1943, 2059, 2117, 2813, 2929, 2987, ..., .
a(n)==61 for n's: 51, 867, 2193, 3009, 3417, 6477, 7089, 8007, 8313, 8517, 10047, 10149, 11577, 11679, ..., .
a(n)==65 for n's: 159, 6837, 8427, 9381, 11289, 12561, 15423, 17331, 23691, 25917, 26553, 30687, 31323, ..., .
a(n)==77 for n's: 41, 287, 533, 697, 1517, 1681, 1927, 2419, 2747, 2911, 3239, 3731, 3977, 4141, 4387, ..., .
(End)

Links

Crossrefs

Cf. A000367, A002445, A180315. [Robert G. Wilson v, Aug 27 2010]

Programs

  • Mathematica
    f[n_] := Block[{b = BernoulliB[2 n]}, Mod[Numerator@b, Denominator@b]]; Array[f, 53, 0] (* Robert G. Wilson v, Aug 27 2010 *)
  • PARI
    a(n) = my(b = bernfrac(2*n)); numerator(b) % denominator(b); \\ Michel Marcus, Mar 15 2015

Formula

A000367(n) mod A002445(n). [Robert G. Wilson v, Aug 27 2010]

A180315 AbsoluteValue(Numerator(Bernoulli(2n))) mod denominator(Bernoulli(2n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 691, 1, 47, 775, 41, 17, 691, 1, 59, 12899, 47, 1, 638653, 1, 2011, 1, 53, 41, 14477, 5, 83, 775, 59, 53, 22298681, 1, 47, 62483, 1, 289, 48540859, 1, 1, 37, 47717, 77, 1058237, 1, 5407, 230759, 77, 1, 1450679, 1, 4471, 61, 83, 101, 71532367
Offset: 0

Views

Author

Robert G. Wilson v, Aug 27 2010

Keywords

Comments

From the von Staudt-Clausen theorem, denominator(B_2n) = product of primes p such that (p-1)|2n.
Values sorted: 1, 5, 17, 37, 41, 47, 49, 53, 59, 61, 65, 77, 83, 89, 101, 113, 137, 161, 167, 169, 173, ..., .
a(n).==1 for n's: 1, 2, 3, 4, 7, 13, 17, 19, 21, 31, 34, 37, 38, 43, 47, 49, 57, 59, 61, 62, 67, 71, 73, ..., .
a(n).==5 for n's: 5, 25, 85, 185, 235, 295, 305, 335, 355, 365, 395, 425, 505, 535, 635, 685, 695, ..., .A051229
a(n)==17 for n's: 11, 77, 87, 121, 143, 187, 407, 517, 539, 649, 671, 737, 781, 847, 869, 1067, 1111, ..., .
a(n)==37 for n's: 39, 507, 1209, 1677, 3783, 4251, 5421, 5811, 6123, 6357, 6513, 7527, 7683, 7761, 8229, ..., .
a(n)==41 for n's: 10, 23, 123, 161, 170, 391, 437, 529, 610, 710, 851, 1010, 1081, 1127, 1357, 1403, ..., .
a(n)==47 for n's: 8, 16, 32, 64, 152, 248, 304, 376, 472, 496, 752, 824, 872, 992, 1256, 1336, 1504, ..., .
a(n)==49 for n's: 55, 275, 605, 2035, 3025, 3355, 3685, 3905, 4345, 5555, 5885, 6985, 7535, 7645, 8195, ..., .
a(n)==53 for n's: 22, 29, 203, 242, 374, 377, 493, 841, 899, 1073, 1247, 1298, 1342, 1363, 1562, 1711, ..., .
a(n)==59 for n's: 14, 28, 266, 434, 532, 868, 994, 1414, 1442, 1526, 1918, 2534, 2758, 2884, 2954, 3052, ..., .
a(n)==61 for n's: 51, 867, 2193, 3009, 3417, 6477, 7089, 8007, 8313, 8517, 10047, 10149, 11577, 11679, ..., .
a(n)==65 for n's: 159, 6837, 8427, 9381, 11289, 12561, 15423, 17331, 23691, 25917, 26553, 30687, 31323, ..., .
a(n)==77 for n's: 41, 46, 92, 287, 533, 697, 782, 874, 1058, 1517, 1681, 1748, 1927, 2116, 2162, 2419, ..., .
a(n)==83 for n's: 26, 52, 494, 988, 1534, 1586, 2626, 2678, 2782, 3068, 3172, 3562, 3874, 4082, 4342, ..., .
a(n)==89 for n's: 58, 1682, 1798, 2726, 3422, 5974, 7946, 8642, 8758, 9106, 12934, 13166, 13282, 13978, ..., .

Links

Crossrefs

Cf. A000367, A002445, A169980.

Programs

  • Mathematica
    f[n_] := Block[{b = BernoulliB[2 n]}, Mod[Abs@ Numerator@ b, Denominator@ b]]; Array[f, 53, 0]

Formula

|A000367(n)| mod A002445(n).

A081863 Largest integer m such that m divides (sigma_(2n+1)(2k-1)-sigma(2k-1)) for all k>=1.

Original entry on oeis.org

24, 240, 168, 480, 264, 21840, 24, 16320, 3192, 2640, 552, 43680, 24, 6960, 57288, 32640, 24, 15353520, 24, 216480, 7224, 5520, 1128, 1485120, 264, 12720, 3192, 13920, 1416, 454293840, 24, 65280, 258888, 240, 18744, 2241613920, 24, 240, 13272, 7360320, 1992
Offset: 1

Views

Author

Benoit Cloitre, Apr 12 2003

Keywords

Comments

a(n)==0 mod 24. It seems that a(n)==0 (mod 2n+1) if and only if 2n+1 is an odd prime.
It appears that a(n)=24 for n in A045979, a(n)=168 for n in A051227, a(n)=264 for n in A051229, and a(n)=240 or 480 if n is in A051225. - Michel Marcus, Dec 07 2013

Crossrefs

Cf. A000203.

Programs

  • PARI
    ds(n, k) = sigma(2*k-1, 2*n+1) - sigma(2*k-1);
a(n) = {my(m = ds(n, 1)); for (k=2, 100, m = gcd(m, ds(n, k));); m;} \\ Script computes gcd of 100 terms; for current data, 10 terms are actually sufficient; is there a better way? - Michel Marcus, Dec 07 2013

Extensions

a(12) corrected and more terms from Michel Marcus, Dec 07 2013
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