A045979 -id:A045979 - cp's OEIS frontend

A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

2, 14, 26, 34, 38, 62, 74, 86, 94, 98, 118, 122, 134, 142, 146, 158, 182, 194, 202, 206, 214, 218, 254, 266, 274, 278, 298, 302, 314, 326, 334, 338, 362, 386, 394, 398, 422, 434, 446, 454, 458, 482, 494, 514, 518, 526, 538, 542, 554, 566, 578

Offset: 1

N. J. A. Sloane

Alternative definition: let D(m) = set of divisors of m; sequence gives n such that the set 1 + D(n) contains only two primes, 2 and 3. E.g., n=98: D(98)={1,2,7,15,49,98}, 1+D = {2,3,8,16,50,99} of which only 2 terms are prime numbers: {2,3}. Observation by Labos Elemer, Jun 24 2002. This is a consequence of the von Staudt-Clausen theorem. - N. J. A. Sloane, Jan 04 2004
The fraction of Bernoulli numbers with denominator 6 is roughly 1/6, see Erdős-Wagstaff. But calculations by H. Cohen and G. Tenenbaum suggest that the fraction is closer to 1/7 (posting to Number Theory List around Dec 20 2005).
Simon Plouffe reports (Feb 13 2007) that at B_{9083002} the proportion is 0.151848915149418661363281... and still decreasing very slowly.
In his PhD thesis at the University of Illinois (see reference), Richard Sunseri proved that a higher proportion of Bernoulli denominators equal 6 than any other value.
Rado showed that for a given Bernoulli number B_n there exist infinitely many Bernoulli numbers B_m having the same denominator. As a special case, if n = 2p where p is an odd prime p == 1 (mod 3), then the denominator of the Bernoulli number B_n equals 6. - Bernd C. Kellner, Mar 21 2018
Conjecture: When the expression (p+q^b)/2 is required to be prime, p is prime, and q is a prime >=5, then all p values are prime congruent to 1 (mod 12) (A068228), if and only if the exponent b is a member of this set. - Richard R. Forberg, Apr 07 2025
There are additional exponential expressions conjectured for generating each of several known prime subsequences (e.g., Pythagorean primes, A002144) where the sequence is invariant to the exponent, if and only if the exponent is a member of this set.  See Forberg link. - Richard R. Forberg, Apr 25 2025

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
C. J. Moreno and S. S. Wagstaff, Sums of Squares of Integers, CRC Press, 2005, Sect. 3.9.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 10.

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Erdős and Samuel S. Wagstaff, Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), pp. 104-112, MR 81c:10064.
Richard R. Forberg, A051222 and Exponential Expressions the Reproduce Certain Prime Subsets, 2025.
K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.
R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 88-90.
Richard Sunseri, Zeros of p-adic L-functions and densities relating to Bernoulli numbers, PhD thesis, University of Illinois, 1979.
Index entries for sequences related to Bernoulli numbers.

Except for 2, all terms are even nontotient numbers. Proper subset of A005277: e.g., 50 and 90 are not here. - Labos Elemer
A112772 is a subsequence. - Bernd C. Kellner, Mar 21 2018
Cf. A045979, A000005, A067513, A002202, A005277.

di[x_] := Divisors[x]
dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1
Do[s=Length[dp[n]]; If[Equal[s, 2], Print[n]], {n, 1, 10000}] (* Labos Elemer *)
Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] (* Labos Elemer *)
Do[s=1+Divisors[n];s1=Flatten[Position[PrimeQ[s], True]]; (*analogous [suitably modified] pairs of programs yield A051225-A051230*) s2=Part[s, s1];If[Equal[s2, {2, 3}], Print[n]], {n, 1, 100}] (* Labos Elemer *)
Select[Range[600],Denominator[BernoulliB[#]]==6&] (* Harvey P. Dale, Dec 08 2011 *)

for(n=1,10^3,if(denominator(bernfrac(n))==6,print1(n,", "))); \\ Joerg Arndt, Oct 28 2014

is(n)=if(n%2,return(0)); fordiv(n/2,d,if(isprime(2*d+1)&&d>1, return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2014

Additional comments and references from Sam Wagstaff, Dec 20 2005

A051225 Numbers m such that the Bernoulli number B_{2*m} has denominator 30.

Original entry on oeis.org

2, 4, 34, 38, 62, 76, 94, 118, 122, 124, 142, 188, 202, 206, 214, 218, 236, 244, 274, 298, 302, 314, 334, 362, 394, 412, 422, 436, 446, 454, 458, 482, 514, 526, 538, 542, 566, 578, 604, 622, 626, 628, 634, 662, 668, 674, 694, 698, 706, 722, 724, 734, 758

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

			The numbers m = 2, 4, 34 are in the list because B_4 = B_8 = -1/30 and B_68 = -78773130858718728141909149208474606244347001/30. - _Petros Hadjicostas_, Jun 06 2020

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Von Staudt-Clausen theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A045979, A051222, A051226, A051227, A051228, A051229, A051230.

Cases[Range[760], n_ /; Denominator[BernoulliB[2*n]] == 30] (* Jean-François Alcover, Mar 23 2011 *)

is(n)=fordiv(n,d, if(isprime(2*d+1) && d>2, return(0))); n%2==0 \\ Charles R Greathouse IV, Jun 21 2017

@p=(2,3,5); $p=5; for($n=4; $n<=1516; $n+=4){while($p<$n+1){$p+=2; next if grep$p%$==0,@p; push@p,$p; push@c,$p-1; }print$n/2,","if!grep$n%$==0,@c; }print"\n"

a(n) = A051226(n)/2. - Petros Hadjicostas, Jun 06 2020

More terms and Perl program from Hugo van der Sanden

Name edited by Petros Hadjicostas, Jun 06 2020

A051230 Numbers m such that the Bernoulli number B_m has denominator 66.

Original entry on oeis.org

10, 50, 170, 370, 470, 590, 610, 670, 710, 730, 790, 850, 1010, 1070, 1270, 1370, 1390, 1490, 1630, 1670, 1850, 1970, 1990, 2230, 2270, 2290, 2570, 2630, 2690, 2770, 2830, 2890, 2950, 3050, 3070, 3110, 3130, 3170, 3310, 3350, 3470, 3530

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

			The numbers m = 10, 50 belong to the list because B_10 = 5/66 and B_50 = 495057205241079648212477525/66. - _Petros Hadjicostas_, Jun 06 2020

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Von Staudt-Clausen theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A045979, A051222, A051225, A051226, A051227, A051228.

Equals 2*A051229.

denoBn[n_?EvenQ] := Times @@ Select[Prime /@ Range[PrimePi[n] + 1], Divisible[n, # - 1] & ]; Select[ Range[10, 4000, 10], denoBn[#] == 66 &] (* Jean-François Alcover, Jun 27 2012, after comments *)
Flatten[Position[BernoulliB[Range[4000]],?(Denominator[#]==66&)]] (* _Harvey P. Dale, Nov 17 2014 *)

/* define indicator function */ a(n)=local(s); s=0; fordiv(n,d,s+=isprime(d+1)&(d>2)&(d!=10)); !s /* get sequence */ an=vector(45,n,0); m=0; forstep(n=10,4000,10, if(a(n),an[ m++ ]=n)); for(n=1,42,print1(an[ n ]","))

More terms from Michael Somos

Name edited by Petros Hadjicostas, Jun 06 2020

A051229 Numbers m such that the Bernoulli number B_{2*m} has denominator 66.

Original entry on oeis.org

5, 25, 85, 185, 235, 295, 305, 335, 355, 365, 395, 425, 505, 535, 635, 685, 695, 745, 815, 835, 925, 985, 995, 1115, 1135, 1145, 1285, 1315, 1345, 1385, 1415, 1445, 1475, 1525, 1535, 1555, 1565, 1585, 1655, 1675, 1735, 1765

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

			The numbers m = 5, 25 belong to the list because B_10 = 5/66 and B_50 = 495057205241079648212477525/66. - _Petros Hadjicostas_, Jun 06 2020

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Von Staudt-Clausen theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A045979, A051222, A051225, A051226, A051227, A051228.

Equals A051230/2.

Select[Range[2000],Denominator[BernoulliB[2 #]]==66&] (* Harvey P. Dale, Mar 11 2012 *)

is(n)=denominator(bernfrac(2*n))==66 \\ Charles R Greathouse IV, Feb 06 2017

[n for n in (1..2000) if denominator(bernoulli(2*n))==66 ] # G. C. Greubel, Jun 06 2020

a(n) = 5*A119456(n). - G. C. Greubel, Jun 06 2020

More terms from Michael Somos

Name edited by Petros Hadjicostas, Jun 06 2020

A051226 Numbers m such that the Bernoulli number B_m has denominator 30.

Original entry on oeis.org

4, 8, 68, 76, 124, 152, 188, 236, 244, 248, 284, 376, 404, 412, 428, 436, 472, 488, 548, 596, 604, 628, 668, 724, 788, 824, 844, 872, 892, 908, 916, 964, 1028, 1052, 1076, 1084, 1132, 1156, 1208, 1244, 1252, 1256, 1268, 1324, 1336, 1348, 1388

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

			The numbers m = 4, 8, 68 are in the list because B_4 = B_8 = -1/30 and B_68 = -78773130858718728141909149208474606244347001/30. - _Petros Hadjicostas_, Jun 06 2020

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Von Staudt-Clausen theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A045979, A051222, A051225, A051227, A051228, A051229, A051230.

Select[Table[n, {n, 4, 1500, 2}], Denominator @ BernoulliB[#] == 30 &] [[1 ;; 47]] (* Jean-François Alcover, Apr 08 2011 *)

lista(nn) = for (n=1, nn, if (denominator(bernfrac(n)) == 30, print1(n, ", "))); \\ Michel Marcus, Mar 30 2015

@p=(2,3,5); $p=5; for($n=4; $n<=1388; $n+=4){while($p<$n+1){$p+=2; next if grep$p%$==0,@p; push@p,$p; push@c,$p-1; }print"$n,"if!grep$n%$==0,@c; }print"\n"

a(n) = 2*A051225(n). - Petros Hadjicostas, Jun 06 2020

More terms and Perl program from Hugo van der Sanden

Name edited by Petros Hadjicostas, Jun 06 2020

A051227 Numbers m such that the Bernoulli number B_{2*m} has denominator 42.

Original entry on oeis.org

3, 57, 93, 129, 177, 201, 213, 237, 291, 327, 381, 417, 447, 471, 489, 501, 579, 591, 597, 633, 669, 681, 687, 807, 921, 951, 1011, 1047, 1059, 1083, 1137, 1149, 1167, 1203, 1227, 1263, 1299, 1317, 1347, 1371, 1389, 1437, 1461, 1497, 1563, 1569

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Von Staudt-Clausen theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A045979, A051222, A051225, A051226, A051228, A051229, A051230.

Select[Range[1600],Denominator[BernoulliB[2#]]==42&] (* Harvey P. Dale, Nov 24 2011 *)

is(n)=denominator(bernfrac(2*n))==42 \\ Charles R Greathouse IV, Feb 07 2017

@p=(2,3,5,7); @c=(4); $p=7; for($n=6; $n<=3126; $n+=6){while($p<$n+1){$p+=2; next if grep$p%$==0,@p; push@p,$p; push@c,$p-1; }print$n/2,","if!grep$n%$==0,@c; }print"\n"

a(n) = A051228(n)/2. - Petros Hadjicostas, Jun 06 2020

More terms and Perl program from Hugo van der Sanden

Name edited by Petros Hadjicostas, Jun 06 2020

A051228 Numbers m such that the Bernoulli number B_m has denominator 42.

Original entry on oeis.org

6, 114, 186, 258, 354, 402, 426, 474, 582, 654, 762, 834, 894, 942, 978, 1002, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1614, 1842, 1902, 2022, 2094, 2118, 2166, 2274, 2298, 2334, 2406, 2454, 2526, 2598, 2634, 2694, 2742, 2778, 2874, 2922, 2994, 3126

Offset: 1

N. J. A. Sloane

From the von Staudt-Clausen theorem, denominator(B_{2*m}) = product of primes p such that (p-1)|2*m.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Von Staudt-Clausen theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A014117, A045979, A051222, A051225, A051226, A051227, A051229, A051230.

2*Select[Range[2000], Denominator[BernoulliB[2#]] == 42 &](* Jean-François Alcover, Nov 25 2011 *)
Position[BernoulliB[Range[3200]],?(Denominator[#]==42&)]//Flatten (* _Harvey P. Dale, Jul 02 2018 *)

is(n)=denominator(bernfrac(n))==42 \\ Charles R Greathouse IV, Feb 07 2017

@p=(2,3,5,7); @c=(4); $p=7; for($n=6; $n<=3126; $n+=6){while($p<$n+1){$p+=2; next if grep$p%$==0,@p; push@p,$p; push@c,$p-1; }print"$n,"if!grep$n%$==0,@c; }print"\n"

a(n) = 2*A051227(n). - Petros Hadjicostas, Jun 06 2020

More terms and Perl program from Hugo van der Sanden

Name edited by Petros Hadjicostas, Jun 06 2020

A119480 Numbers n such that the Bernoulli number B_{4n} has denominator 30.

Original entry on oeis.org

1, 2, 17, 19, 31, 38, 47, 59, 61, 62, 71, 94, 101, 103, 107, 109, 118, 122, 137, 149, 151, 157, 167, 181, 197, 206, 211, 218, 223, 227, 229, 241, 257, 263, 269, 271, 283, 289, 302, 311, 313, 314, 317, 331, 334, 337, 347, 349, 353, 361, 362, 367, 379

Offset: 1

Alexander Adamchuk, Jul 26 2006

nonn

Most a(n) are primes from A043297(n) except for a(1) = 1 and composite a(n) for n=6,10,12,17,18,26,28,38,39,42,45,50,51, ... a(6) = 38 = 2*19, a(10) = 62 = 2*31, a(12) = 94 = 2*47, a(17) = 118 = 2*59, a(18) = 122 = 2*61, a(26) = 206 = 2*103, a(28) = 218 = 2*109, a(38) = 289 = 17*17, a(39) = 302 = 2*151, a(42) = 314 = 2*157, a(45) = 334 = 2*167, a(50) = 361 = 19*19, a(51) = 362 = 2*181, ... It appears that most composite a(n) are the doubles of some primes from A043297(n) belonging to A081092[n] and A045404[n] - Primes congruent to {3, 4, 5, 6} mod 7. The rest of composite a(n) are the squares of the primes from A043297(n).

Some a(n) are the products of different primes from A043297(n), for example a(77) = 527 = 17*31. a(n) belong to A045402 Primes congruent to {1, 3, 4, 5, 6} mod 7. a(n) is a subset of A053176 Primes p such that 2p+1 is composite, A045979 Bernoulli number B_{2n} has denominator 6, A090863 Numbers n such that F(n+1)*F(n-1)*B(2n) is an integer, where F(k)=k-th Fibonacci number and B(2k)=2k-th Bernoulli number. - Alexander Adamchuk, Jul 27 2006

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

Cf. A043297, A051225, A051222, A051230, A045404.

Cf. A045402, A053176, A045979, A090863.

Select[Range@ 400, Denominator@ BernoulliB[4 #] == 30 &] (* Michael De Vlieger, Aug 09 2017 *)

a(n) = A051225[n]/2.

A271634 Numbers n such that Bernoulli number B_{n} has denominator 510.

Original entry on oeis.org

16, 32, 64, 128, 304, 496, 608, 752, 944, 992, 1504, 1648, 1744, 1984, 2512, 2672, 3008, 3152, 3296, 3376, 3488, 3568, 3632, 3664, 3856, 3968, 4112, 4208, 4528, 4976, 5024, 5072, 5344, 5584, 5648, 5776, 5872, 6016, 6064, 6128, 6224, 6304, 6592, 6752, 7024, 7136, 7264

Offset: 1

Paolo P. Lava, Apr 21 2016

510 = 2 * 3 * 5 * 17.
All terms are multiple of a(1) = 16.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 463.

			Bernoulli B_{16} is -3617/510, hence 16 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, 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(10^6,510);

Select[Range[0, 1000], Denominator[BernoulliB[#]] == 510 &] (* Robert Price, Apr 21 2016 *)

isok(n) = denominator(bernfrac(n)) == 510; \\ Michel Marcus, Apr 22 2016

More terms from Michel Marcus, Apr 22 2016

A271635 Numbers n such that Bernoulli number B_{n} has denominator 138.

Original entry on oeis.org

22, 154, 242, 286, 374, 814, 1034, 1078, 1298, 1342, 1474, 1562, 1694, 1738, 2134, 2222, 2354, 2794, 3014, 3058, 3146, 3278, 3454, 3586, 3674, 3982, 4114, 4246, 4334, 4378, 4906, 4994, 5654, 5698, 5786, 5918, 5962, 6094, 6226, 6754, 6842, 6886, 6974, 7414, 7634, 7678, 7766

Offset: 1

Paolo P. Lava, Apr 21 2016

138 = 2 * 3 * 23.
All terms are multiple of a(1) = 22.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 17.

			Bernoulli B_{22} is 854513/138, hence 22 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, 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(10^6,138);

Select[Range[0, 1000], Denominator[BernoulliB[#]] == 138 &] (* Robert Price, Apr 21 2016 *)

isok(n) = denominator(bernfrac(n)) == 138; \\ Michel Marcus, Apr 22 2016

More terms from Michel Marcus, Apr 22 2016

cp's OEIS Frontend

A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

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A051225 Numbers m such that the Bernoulli number B_{2*m} has denominator 30.

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A051230 Numbers m such that the Bernoulli number B_m has denominator 66.

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A051229 Numbers m such that the Bernoulli number B_{2*m} has denominator 66.

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A051226 Numbers m such that the Bernoulli number B_m has denominator 30.

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A051227 Numbers m such that the Bernoulli number B_{2*m} has denominator 42.

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