A051229 -id:A051229 - cp's OEIS frontend

A119657 Denominator of BernoulliB[10p] divided by 66, where p=Prime[n].

5, 217, 1, 71, 23, 131, 1, 191, 47, 59, 311, 1, 83, 431, 1, 107, 1, 1, 1, 1, 1, 1, 167, 179, 971, 1, 1031, 1, 1091, 227, 1, 263, 1, 1, 1, 1511, 1571, 1, 1, 347, 359, 1811, 383, 1931, 1, 1, 2111

Offset: 1

Alexander Adamchuk, Jul 28 2006

nonn

The only composite in this sequence is a(2) = 217 = 7*31. All other a(n) are equal to 1 (for n=3,7,12,15,17,18,19,20,21,22,26,28,31,33,34,35,38,39,45,46..) or prime: a(1) = 5, all other primes in a(n) belong to A068231[n]: Primes congruent to 11 (mod 12). It appears that every prime from A068231[n] except 11 shows up in a(n) just once.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Cf. A068231, A119456, A051230, A051229, A002445.

Table[Denominator[BernoulliB[10Prime[n]]]/66,{n,1,47}]

a(n) = Denominator[BernoulliB[10Prime[n]]]/66.

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)]);

A295589 Numbers k such that Bernoulli number B_{k} has denominator 33330.

Original entry on oeis.org

100, 1700, 7100, 16700, 22300, 25700, 28300, 31300, 31700, 33100, 35300, 37900, 38300, 38900, 39700, 44900, 45700, 47900, 52100, 56900, 58700, 60700, 66100, 75100, 75700, 78700, 79700, 83900, 85700, 85900, 88100, 90700, 96700, 99100

Offset: 1

Paolo P. Lava, Nov 24 2017

33330= 2*3*5*11*101.
All terms are multiples of a(1) = 100.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 28859.

			Bernoulli B_{100} is
-945980378191221252952274330694937218727028415330669361333856962043113954151972 47711/33330, hence 100 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, 33330);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 11, 101}:
select(filter, [seq(i, i=1..10^5)]);

Select[Range[100,100000,100],Denominator[BernoulliB[#]]==33330&] (* Harvey P. Dale, Aug 05 2022 *)

A295590 Numbers k such that Bernoulli number B_{k} has denominator 46410.

Original entry on oeis.org

48, 10128, 16944, 21072, 25008, 28176, 31056, 33648, 35184, 39696, 42288, 52656, 55824, 59952, 60432, 62448, 71664, 73104, 77808, 78096, 82704, 83568, 84432, 91824, 93648, 98544, 100176, 100272, 102288, 107664, 108912, 110256, 110832, 112368, 114096, 117168, 120144

Offset: 1

Paolo P. Lava, Nov 24 2017

46410 = 2*3*5*7*13*17.
All terms are multiples of a(1) = 48.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 31933.

			46410 = 2*3*5*7*13*17.
Bernoulli B_{48} is -5609403368997817686249127547/46410, hence 48 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(10^6,64722);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 7, 13, 17}:
select(filter, [seq(i, i=1..10^5)]);

Select[48*Range[2600],Denominator[BernoulliB[#]]==46410&] (* Harvey P. Dale, May 17 2020 *)

A295591 Numbers k such that Bernoulli number B_{k} has denominator 61410.

Original entry on oeis.org

88, 968, 5192, 5368, 13816, 15928, 19624, 19976, 22616, 23144, 23848, 24904, 27368, 27544, 27896, 29656, 31064, 33704, 34936, 38632, 40216, 40568, 40744, 45848, 46024, 48136, 49544, 50248, 51656, 53416, 56584, 56936, 57112, 59048, 60808, 61688, 67672, 68024, 71368

Offset: 1

Paolo P. Lava, Nov 24 2017

61410 = 2*3*5*23*89.
All terms are multiples of a(1) = 88.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 56003.

			Bernoulli B_{88} is -1311426488674017507995511424019311843345750275572028644296919890574047/61410 hence 88 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, 61410);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 23, 89}:
select(filter, [seq(i, i=1..10^5)]);

isok(n) = denominator(bernfrac(n)) == 61410; \\ Michel Marcus, Jan 07 2018

A295592 Numbers k such that Bernoulli number B_{k} has denominator 64722.

Original entry on oeis.org

66, 3894, 4686, 5214, 6402, 8382, 9174, 9834, 10362, 10758, 11022, 13134, 14718, 17754, 20262, 20922, 22242, 23034, 23298, 25014, 25278, 25674, 26466, 27786, 28974, 29634, 30162, 31614, 34386, 36102, 37554, 37686, 38742, 39534, 40722, 42438, 44418, 45606, 46266

Offset: 1

Paolo P. Lava, Nov 24 2017

64722 = 2*3*7*23*67.
All terms are multiples of a(1) = 66.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 62483.

			Bernoulli B_{66} is
1472600022126335654051619428551932342241899101/64722, hence 66 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(10^6,64722);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 7, 23, 67}:
select(filter, [seq(i, i=1..10^5)]);

A295593 Numbers k such that Bernoulli number B_{k} has denominator 230010.

Original entry on oeis.org

80, 160, 320, 13360, 17840, 18160, 20560, 25360, 26720, 28240, 30640, 35680, 36320, 36560, 41120, 43280, 45520, 46960, 50720, 52880, 56480, 60080, 61280, 69040, 70960, 71360, 72560, 72640, 79280, 84080, 87760, 91040, 92240, 93040, 93680, 93920, 94480, 97040, 97360

Offset: 1

Paolo P. Lava, Nov 24 2017

230010 = 2*3*5*11*17*41.
All terms are multiples of a(1) = 80.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 182293.

			Bernoulli B_{80} is
-4603784299479457646935574969019046849794257872751288919656867/230010, hence 80 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(10^6,230010);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 11, 17, 41}:
select(filter, [seq(i, i=1..10^5)]);

A295594 Numbers k such that Bernoulli number B_{k} has denominator 272118.

Original entry on oeis.org

90, 14670, 24210, 35010, 40410, 41670, 44910, 46890, 55530, 57870, 60570, 60930, 82710, 83610, 87030, 89730, 98370, 101070, 104670, 106830, 109530, 111330, 113310, 114930, 117090, 117270, 117630, 123570, 128610, 138870, 150030, 152730, 160470, 175590, 178110, 179730

Offset: 1

Paolo P. Lava, Nov 24 2017

272118 = 2*3*7*11*19*31.
All terms are multiples of a(1) = 90.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 230759.

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, 272118);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 7, 11, 19, 31}:
select(filter, [seq(i, i=1..10^5)]);

272118 = 2*3*7*11*19*31.
Bernoulli B_{90} is
1179057279021082799884123351249215083775254949669647116231545215727922535/ 272118 hence 90 is in the sequence.

cp's OEIS Frontend

A119657 Denominator of BernoulliB[10p] divided by 66, where p=Prime[n].

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

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

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

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A295589 Numbers k such that Bernoulli number B_{k} has denominator 33330.

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A295590 Numbers k such that Bernoulli number B_{k} has denominator 46410.

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A295591 Numbers k such that Bernoulli number B_{k} has denominator 61410.

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A295592 Numbers k such that Bernoulli number B_{k} has denominator 64722.