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 14 results. Next

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

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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

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

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(10^6,13530);
# 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)]);
  • Mathematica
    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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

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

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(10^6,14322);
# 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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

			Bernoulli B_{100} is
-945980378191221252952274330694937218727028415330669361333856962043113954151972 47711/33330, hence 100 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.

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, 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)]);
  • Mathematica
    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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

			46410 = 2*3*5*7*13*17.
Bernoulli B_{48} is -5609403368997817686249127547/46410, hence 48 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.

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(10^6,64722);
# 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)]);
  • Mathematica
    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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

			Bernoulli B_{88} is -1311426488674017507995511424019311843345750275572028644296919890574047/61410 hence 88 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.

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, 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)]);
  • PARI
    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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

			Bernoulli B_{66} is
1472600022126335654051619428551932342241899101/64722, hence 66 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.

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(10^6,64722);
# 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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

Examples

			Bernoulli B_{80} is
-4603784299479457646935574969019046849794257872751288919656867/230010, hence 80 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.

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(10^6,230010);
# 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

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

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.

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.

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, 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)]);

Formula

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

A295596 Numbers k such that Bernoulli number B_{k} has denominator 3404310.

Original entry on oeis.org

84, 168, 16548, 26628, 29316, 38388, 43764, 47964, 53256, 61572, 69132, 71988, 72156, 73668, 87528, 96852, 103908, 109284, 121548, 123144, 124572, 137508, 139188, 142548, 144312, 144564, 146244, 147336, 156828, 163716, 167748, 172452, 174972, 185388, 188076, 190428
Offset: 1

Views

Author

Paolo P. Lava, Nov 24 2017

Keywords

Comments

3404310 = 2*3*5*7*13*29*43.
All terms are multiples of a(1) = 84.
For these numbers numerator(B_{k}) mod denominator(B_{k}) = 2346073.

Examples

			Bernoulli B_{84} is
-2024576195935290360231131160111731009989917391198090877281083932477/3404310 hence 84 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.

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, 3404310);
# Alternative: # according to Robert Israel code in A282773
with(numtheory): filter:= n ->
select(isprime, map(`+`, divisors(n), 1)) = {2, 3, 5, 7, 13, 29, 43}:
select(filter, [seq(i, i=1..10^5)]);
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