A051227 -id:A051227 - cp's OEIS frontend

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

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

Paolo P. Lava, Apr 22 2016

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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

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);

Select[20 Range@ 850, Denominator@ BernoulliB@ # == 330 &] (* Michael De Vlieger, Apr 29 2016 *)

isok(n) = denominator(bernfrac(n)) == 330; \\ Michel Marcus, Apr 22 2016

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

Paolo P. Lava, Apr 22 2016

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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

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);

Select[46 Range@ 320, Denominator@ BernoulliB@ # == 282 &] (* Michael De Vlieger, Apr 29 2016 *)

isok(n) = denominator(bernfrac(n)) == 282; \\ Michel Marcus, Apr 22 2016

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

Paolo P. Lava, Apr 22 2016

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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

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);

Select[28 Range@ 520, Denominator@ BernoulliB@ # == 870 &] (* Michael De Vlieger, Apr 29 2016 *)

isok(n) = denominator(bernfrac(n)) == 870; \\ Michel Marcus, Apr 22 2016

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

Paolo P. Lava, Apr 22 2016

nonn

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

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

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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

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);

isok(n) = denominator(bernfrac(n)) == 690; \\ Michel Marcus, Apr 22 2016

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

Paolo P. Lava, Apr 28 2016

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

			Bernoulli B_{92} is -1295585948207537527989427828538576749659341483719435143023316326829946247/1410, hence 92 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

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);

Select[92 Range@ 360, Denominator@ BernoulliB@ # == 1410 &] (* Michael De Vlieger, Apr 28 2016 *)

lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 1410, print1(n, ", "))); \\ Altug Alkan, Apr 28 2016

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

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

Paolo P. Lava, Mar 07 2017

nonn

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

			Bernoulli B_{82} is 1677014149185145836823154509786269900207736027570253414881613/498, hence 82 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Von Staudt-Clausen theorem

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.

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

Select[82 Range[360], Denominator@ BernoulliB@ # == 498 &] (* Michael De Vlieger, Mar 07 2017 *)

More terms from Michael De Vlieger, Mar 07 2017

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

Benoit Cloitre, Apr 12 2003

nonn

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

Cf. A000203.

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

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

A272383 Numbers n such that Bernoulli number B_{n} has denominator 3318.

Original entry on oeis.org

78, 1014, 2418, 3354, 7566, 8502, 10842, 11622, 12246, 12714, 13026, 15054, 15366, 15522, 16458, 17394, 23946, 26286, 27222, 27534, 29562, 29874, 30342, 31434, 31902, 33774, 34242, 35646, 36114, 40794, 42198, 43602, 44538, 47814, 48126, 48282, 49218, 50154, 52494, 55302, 57174, 57642, 59046, 59982

Offset: 1

Paolo P. Lava, Apr 28 2016

3318 = 2 * 3 * 7 * 79.
All terms are multiples of a(1) = 78.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 37.

			Bernoulli B_{78} is 414846365575400828295179035549542073492199375372400483487/3318, hence 78 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

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(10^6,3318);

Select[78 Range@ 800, Denominator@ BernoulliB@ # == 3318 &] (* Michael De Vlieger, Apr 28 2016 *)

lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 3318, print1(n, ", "))); \\ Altug Alkan, Apr 28 2016

from sympy import divisors, isprime
A272383_list = []
for i in range(78, 10**6, 78):
    for d in divisors(i):
        if d not in (1,2,6,78) and isprime(d+1):
            break
    else:
        A272383_list.append(i) # Chai Wah Wu, May 02 2016

a(9)-a(22) from Altug Alkan, Apr 28 2016

More terms from Michael De Vlieger, Apr 28 2016

A295587 Numbers k such that Bernoulli number B_{k} has denominator 13530.

Original entry on oeis.org

40, 6680, 7880, 8920, 9080, 10280, 12520, 12680, 14120, 15320, 15560, 18280, 20840, 21640, 22760, 23480, 25720, 26440, 28040, 30040, 30280, 31880, 33080, 33560, 34520, 35240, 35480, 36280, 38680, 39640, 42040, 43880, 44360, 46120, 46520, 46840, 47240, 47720, 48520

Offset: 1

Paolo P. Lava, Nov 24 2017

13530 = 2*3*5*11*41.
All terms are multiples of a(1) = 40.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 11519.

			Bernoulli B_{40} is -261082718496449122051/13530, hence 40 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186, A272369.

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(10^6,13530);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 11, 41}:
select(filter, [seq(i, i=1..10^5)]);

Select[Range[50000],Denominator[BernoulliB[#]]==13530&] (* Harvey P. Dale, Jul 29 2025 *)

A295588 Numbers k such that Bernoulli number B_{k} has denominator 14322.

Original entry on oeis.org

30, 1770, 3810, 4170, 4470, 4890, 5910, 5970, 6810, 8070, 9210, 10590, 11370, 11670, 12030, 12990, 13470, 13890, 14370, 14970, 15630, 16890, 17070, 17610, 18510, 18570, 19290, 19410, 20190, 20310, 21270, 22710, 24810, 25710, 26310, 27570, 27870, 29010, 29490, 29730

Offset: 1

Paolo P. Lava, Nov 24 2017

14322 = 2*3*7*11*31.
All terms are multiples of a(1) = 30.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 12899.

			Bernoulli B_{30} is 8615841276005/14322, hence 30 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A045979, A051222, A051225, A051226, A051227, A051228, A051229, A051230, A119456, A119480, A249134, A255684, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186, A272369.

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(10^6,14322);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 7, 11, 31}:
select(filter, [seq(i, i=1..10^5)]);

cp's OEIS Frontend

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

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

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

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

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

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

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Maple

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

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