A051229 -id:A051229 - cp's OEIS frontend

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

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

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

A119456 Numbers m such that the Bernoulli number B_{10*m} has denominator 66.

Original entry on oeis.org

1, 5, 17, 37, 47, 59, 61, 67, 71, 73, 79, 85, 101, 107, 127, 137, 139, 149, 163, 167, 185, 197, 199, 223, 227, 229, 257, 263, 269, 277, 283, 289, 295, 305, 307, 311, 313, 317, 331, 335, 347, 353, 355, 365, 373, 379, 383, 389, 395, 397, 401, 433, 449, 457, 461

Offset: 1

Alexander Adamchuk, Jul 26 2006

nonn

Subset of A002181 (inverse of the Euler totient function).
Most terms are primes except for n = 12, 21, 32, 33, 34, 40, ... because a(12) = 85 = 5*17, a(21) = 185 = 5*37, a(32) = 289 = 17*17, a(33) = 295 = 5*59, a(34) = 305 = 5*61, a(40) = 335 = 5*67, ... Each composite term appears to be a product of two primes from previous terms or a square of a prime from previous terms.
Composite terms are the products of powers of primes that are factors of previous terms. For example, there are terms equal to 17, 17^2, 5*17^2, 59^2, 59*61, 61^2, 61*67, 67^2, 67*73, 17^3, 5*17*59, 71*73, 5*17*61, 73^2, 71*79, 73*79, 5*17*73, 79^2, 61*167, 101^2, 37*277, 5*37*59, 79*139, 107^2, 5*17*139, 5*37*67, 5*37*71, 17^2*47, 61*223, 61*227, 5*17*163, 5*17*167, 71*227, 127^2, 17^2*59, 5*59^2, 17^2*61, 5*61^2, 137^2, 137*139, 139^2, 17^2*67, 5*17*229, 137*149, 5*61*67, 5*59*71, 17^2*73, 5*67^2, 5*61*79, 5*67*73, 5*17^3, ... - Alexander Adamchuk, Jul 28 2006

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

Cf. A002181, A002445, A051229, A051230.

Do[s=1+Divisors[n]; s1=Flatten[Position[PrimeQ[s], True]]; s2=Part[s, s1]; If[Equal[s2, {2, 3, 11}], Print[n/10]], {n, 1, 50000}] (* Alexander Adamchuk, Jul 28 2006 *)

isok(m) = denominator(bernfrac(10*m)) == 66; \\ Michel Marcus, May 31 2022

a(n) = A051230(n)/10 = A051229(n)/5.

More terms from Alexander Adamchuk, Jul 28 2006

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

A272138 Numbers n such that Bernoulli number B_{n} has denominator 798.

Original entry on oeis.org

18, 54, 342, 558, 774, 1026, 1206, 1674, 1962, 2322, 2826, 2934, 3006, 3474, 3618, 3798, 4014, 4086, 4122, 4842, 5706, 5886, 6282, 6354, 6498, 6894, 7002, 7362, 7578, 7794, 7902, 8082, 8226, 8334, 8478, 8766, 8982, 9018, 9378, 9414, 9846, 10134, 10278, 10422, 10602, 10782

Offset: 1

Paolo P. Lava, Apr 21 2016

798 = 2 * 3 * 7 * 19.
All terms are multiple of a(1) = 18.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 775.

			Bernoulli B_{18} is 43867/798, hence 18 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, 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(10^6,798);

Select[Range[0, 1000], Denominator[BernoulliB[#]] == 798 &] (* Robert Price, Apr 21 2016 *)

lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 798, print1(n, ", "))); \\ Altug Alkan, Apr 22 2016

More terms from Altug Alkan, Apr 22 2016

A272139 Numbers n such that Bernoulli number B_{n} has denominator 1806.

Original entry on oeis.org

42, 294, 798, 1806, 2058, 2814, 2982, 4074, 4578, 5334, 5586, 6594, 6846, 8106, 8274, 8358, 9366, 9534, 12642, 12894, 13314, 14154, 14658, 15162, 17178, 18186, 19194, 20118, 20454, 21882, 21966, 22722, 22974, 23982, 25914, 26502, 27006, 28266, 28518, 29778

Offset: 1

Paolo P. Lava, Apr 21 2016

nonn

1806 = 2 * 3 * 7 * 43.
All terms are multiple of a(1) = 42.
For these numbers numerator(B_{n}) mod denominator(B_{n}) = 1.
In 2005, B. C. Kellner proved E. W. Weisstein's conjecture that denom(B_n) = n only if n = 1806.

			Bernoulli B_{42} is 1520097643918070802691/1806, hence 42 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, 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(10^6,1806);

Select[Range[0, 1000], Denominator[BernoulliB[#]] == 1806 &] (* Robert Price, Apr 21 2016 *)
Select[Range[42,30000,42],Denominator[BernoulliB[#]]==1806&] (* Harvey P. Dale, Jun 01 2019 *)

lista(nn) = for(n=1, nn, if(denominator(bernfrac(n)) == 1806, print1(n, ", "))); \\ Altug Alkan, Apr 22 2016

More terms from Altug Alkan, Apr 22 2016

cp's OEIS Frontend

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

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A119456 Numbers m such that the Bernoulli number B_{10*m} has denominator 66.

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