A051230 -id:A051230 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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.

A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

Original entry on oeis.org

1, 2, 6, 1, 30, 2, 42, 1, 30, 2, 66, 1, 2730, 2, 6, 1, 510, 2, 798, 1, 330, 2, 138, 1, 2730, 2, 6, 1, 870, 2, 14322, 1, 510, 2, 6, 1, 1919190, 2, 6, 1, 13530, 2, 1806, 1, 690, 2, 282, 1, 46410, 2, 66, 1, 1590, 2, 798, 1, 870, 2, 354, 1

Offset: 0

Paul Curtz, Oct 28 2014

nonn

There exist an infinity of 1's, 2's, 6's, 30's, 42's, 66's, ... .
Respective ranks:
   0,   3,    7,   11,   15,   19, ...
   1,   5,    9,   13,   17,   21, ...   (= A016813)
   2,  14,   26,   34,   38,   62, ...   (= A051222)
   4,   8,   68,   76,  124,  152, ...   (= A051226)
   6, 114,  186,  258,  354,  402, ...   (= A051228)
  10,  50,  170,  370,  470,  590, ...   (= A051230)
  12,  24, 1308, 1884, 2004, 2364, ...   (= A249134)
  etc.
Hence by antidiagonals a permutation of A001477(n).
First column: A248614(n).
a(n) is an alternative sequence for the denominators of the Bernoulli numbers.
First 36 terms of the corresponding clockwise spiral:
.
  330------2----138------1---2730------2
    |                                  |
    |                                  |
    1     42------1-----30------2      6
    |      |                    |      |
    |      |                    |      |
  798      2      1------2     66      1
    |      |             |      |      |
    |      |             |      |      |
    2     30------1------6      1    870
    |                           |      |
    |                           |      |
  510------1------6------2---2730      2
                                       |
                                       |
    1------6------2----510------1--14322

Index entries for sequences related to Bernoulli numbers

A variant of the Clausen numbers A141056, A160014. And of A176591.

Cf. A000034, A002445, A016813, A027642, A051222, A051226, A051228, A051230, A090126, A164020, A248614, A249134.

Clausen := proc(n) local S, i;
S := numtheory[divisors](n); S := map(i->i+1, S);
S := select(isprime, S); mul(i, i=S) end:
A249306 := n -> `if`(n mod 4 = 3, 1, Clausen(n)):
seq(A249306(n), n=0..59); # Peter Luschny, Nov 10 2014

a[n_] := Denominator[BernoulliB[n]]; a[n_ /; Mod[n, 4] == 1] = 2; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2014 *)

a(2n) = A002445(n), a(2n+1) = A000034(n+1).

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)]);

cp's OEIS Frontend

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

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

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A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

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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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