A051225 -id:A051225 - cp's OEIS frontend

A043297 Primes p such that B(4*p) has denominator 30 where B(2n) are the Bernoulli numbers.

2, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, 211, 223, 227, 229, 241, 257, 263, 269, 271, 283, 311, 313, 317, 331, 337, 347, 349, 353, 367, 379, 383, 389, 397, 401, 421, 439, 449, 457, 461, 463, 467, 479, 503, 521

Offset: 1

Benoit Cloitre, Mar 24 2002

Complement of A087634, primes p such that phi(k) = 4p has a solution, where phi is Euler's totient function.
The sequences a(n), A005384 and A023212 form a partition of the set of primes > 3: Using von Staudt-Clausen formula the divisors of 4p increased by 1 are {2,3,5,p+1,2p+1,4p+1}, p+1 is clearly an even number, and if 2p+1 and 4p+1 are not prime, then denom(B(4p))=30. - Enrique Pérez Herrero, Aug 15 2011
Also 2 with the primes p such that both 2*p+1 and 4*p+1 are composite: A210684. For these numbers k > 2 the equation: phi(n)=k*tau(n), where phi is A000010 and tau is A000005, has no solutions: A112954(a(n))=0. - Enrique Pérez Herrero, May 12 2012

E. Pérez Herrero, Table of n, a(n) for n=1..30000
Wikipedia, Von Staudt-Clausen theorem

Cf. A051225, A053176, A087634.
Cf. A005384, A023212.
Cf. A210684.

Select[Prime[Range[100]], Denominator[BernoulliB[4# ]]==30&] (* T. D. Noe, Feb 19 2004 *)
Select[Prime[Range[100]],!PrimeQ[4#+1]&&!PrimeQ[2#+1]||(#==2)&] (* Enrique Pérez Herrero, Aug 16 2011 *)

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

Robert G. Wilson v, Aug 19 2010

nonn

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)

Robert G. Wilson v, Table of n, a(n) for n = 0..50000.
Index entries for sequences related to Bernoulli numbers. [From _Robert G. Wilson v_, Aug 27 2010]

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

f[n_] := Block[{b = BernoulliB[2 n]}, Mod[Numerator@b, Denominator@b]]; Array[f, 53, 0] (* Robert G. Wilson v, Aug 27 2010 *)

a(n) = my(b = bernfrac(2*n)); numerator(b) % denominator(b); \\ Michel Marcus, Mar 15 2015

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

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

A043298 Numbers n such that B(6*n) has denominator 42 where B(2k) are the Bernoulli numbers.

Original entry on oeis.org

1, 19, 31, 43, 59, 67, 71, 79, 97, 109, 127, 139, 149, 157, 163, 167, 193, 197, 199, 211, 223, 227, 229, 269, 307, 317, 337, 349, 353, 361, 379, 383, 389, 401, 409, 421, 433, 439, 449, 457, 463, 479, 487, 499, 521, 523, 541, 547, 563, 569, 571, 587, 589, 599

Offset: 1

Benoit Cloitre, Mar 24 2002

Except for 1 and 361=19^2 terms listed are primes.

Most a(n) are primes p such that 2p+1 is composite A053176. Nonprime a(n) (except a(1) = 1) are the powers or the products of primes from a(n). For example, 361 = 19^2, 589 = 19*31, 961 = 31^2, 1333 = 31*43, 1849 = 43^2, 2071 = 19*109, 2077 = 31*67, 2201 = 31*71, 2449 = 31*79, 2537 = 43*59, 2641 = 19*139, 2881 = 43*67, 2983 = 19*157, 3053 = 43*71, 3173 = 19*167, ..., 6859 = 19^3. - Alexander Adamchuk, Jul 28 2006

Enrique Pérez Herrero, Table of n, a(n) for n=1..50000

Cf. A051225, A053176.

Do[s=1+Divisors[n]; s1=Flatten[Position[PrimeQ[s], True]]; s2=Part[s, s1]; If[Equal[s2, {2, 3, 7}], Print[n/6]], {n, 1, 10000}] (* Alexander Adamchuk, Jul 28 2006 *)
Select[Range[600],Denominator[BernoulliB[6#]]==42&] (* Harvey P. Dale, Jan 09 2024 *)

Corrected and extended by Ralf Stephan, Oct 21 2002

More terms from Alexander Adamchuk, Jul 28 2006

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

cp's OEIS Frontend

A043297 Primes p such that B(4*p) has denominator 30 where B(2n) are the Bernoulli numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A043298 Numbers n such that B(6*n) has denominator 42 where B(2k) are the Bernoulli numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments