A051222 -id:A051222 - cp's OEIS frontend

A045979 Bernoulli number B_{2n} has denominator 6.

1, 7, 13, 17, 19, 31, 37, 43, 47, 49, 59, 61, 67, 71, 73, 79, 91, 97, 101, 103, 107, 109, 127, 133, 137, 139, 149, 151, 157, 163, 167, 169, 181, 193, 197, 199, 211, 217, 223, 227, 229, 241, 247, 257, 259, 263, 269, 271, 277, 283, 289

Offset: 1

N. J. A. Sloane

All primes in A053176 are in the sequence. If n is in the sequence, its factorization contains only primes in A053176. - Benoit Cloitre, Oct 19 2002
B(2n) has denominator 6 iff (n^2-1)*B(2n) is an integer. - Benoit Cloitre, Feb 15 2004
Subsequence of A156543. - Reinhard Zumkeller, Feb 10 2009

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 76.

Cf. A051222, A053176, A156543.

[n: n in [0..400] | Denominator(Bernoulli(2*n)) eq 6]; // Vincenzo Librandi, Feb 06 2016

ok[n_] := IntegerQ[(n^2 - 1)*BernoulliB[2n]]; Select[Range[300], ok] (* Jean-François Alcover, Jun 27 2012, after Benoit Cloitre *)
result = {}; Do[count = 0;
Do[If[Not[PrimeQ[2*Divisors[n][[i]] + 1]], count++],
{i, 2, DivisorSigma[0, n]}]; If[count == DivisorSigma[0, n] - 1, AppendTo[result, n]], {n, 1, 10000}]; result  (* Richard R. Forberg, Aug 06 2016 *)
Position[BernoulliB[2 Range[300]],?(Denominator[#]==6&)]//Flatten (* _Harvey P. Dale, Jan 28 2017 *)

isok(n) = denominator(bernfrac(2*n)) == 6; \\ Michel Marcus, Feb 06 2016

a(n) seems to be asymptotic to c*n, 5 < c < 6. - Benoit Cloitre, Oct 19 2002

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

A249134 Numbers k such that Bernoulli number B_k has denominator 2730.

Original entry on oeis.org

12, 24, 1308, 1884, 2004, 2364, 2532, 2724, 3804, 4008, 4044, 4188, 4236, 4668, 5052, 5064, 5268, 5388, 5484, 6252, 6492, 6564, 6756, 6852, 7044, 7188, 7356, 7404, 7608, 7764, 8124, 8412, 8472, 8796, 9084, 9228, 9852, 9876, 9924

Offset: 1

Jean-François Alcover and Paul Curtz, Oct 22 2014

nonn

2730 = 2 * 3 * 5 * 7 * 13.

			BernoulliB(12) is -691/2730, hence 12 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000367, A002445, A006954, A027642, A027760, A027762, A051222, A051225-A051230, A245056, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

Reap[For[n = 0, n <= 10^4, n = n+12, If[Denominator[BernoulliB[n]] == 2730, Print[n]; Sow[n]]]][[2, 1]]
Select[Table[n, {n, 2, 10000}], Denominator@BernoulliB[#]==2730 &] (* Vincenzo Librandi, Apr 02 2015 *)

is(n)=denominator(bernfrac(n))==2730 \\ Charles R Greathouse IV, Oct 22 2014

is(n)=if(n%12 || n%16==0 || n%9==0, return(0)); forprime(p=5,107, if(n%p==0, return(0))); fordiv(n,d, if(isprime(d+1) && d>13, return(0))); 1 \\ Charles R Greathouse IV, Oct 22 2014

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.

A067513 Number of divisors d of n such that d+1 is prime.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 2, 1, 4, 1, 4, 1, 4, 1, 3, 1, 5, 1, 2, 1, 4, 1, 5, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 4, 1, 4, 1, 3, 1, 8, 1, 2, 1, 4, 1, 5, 1, 3, 1, 4, 1, 8, 1, 2, 1, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 2, 1, 5, 1, 6, 1, 4, 1, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1

Offset: 1

Amarnath Murthy, Feb 12 2002

1, 2 and 4 are the only numbers such that for every divisor d, d+1 is a prime.
These and only these primes appear as prime divisors of any term of InvPhi(n) set if n is not empty, i.e., if n is from A002202. - Labos Elemer, Jun 24 2002
a(n) is the number of integers k such that n = k - k/p where p is one of the prime divisors of k. (See, e.g., A064097 and A333123, which are related to the mapping k -> k - k/p.) - Robert G. Wilson v, Jun 12 2022

			a(12) = 5 as the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the corresponding primes are 2,3,5,7 and 13. Only 3+1 = 4 is not a prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty, arXiv:2208.06704 [math.NT], 2022.
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty II, arXiv:2209.01087 [math.NT], 2022-2023.
Steve Fan and Carl Pomerance, Shifted-prime divisors, arXiv:2401.10427 [math.NT], 2024.
Mikhail R. Gabdullin, Moments of the shifted prime divisor function, arXiv preprint (2025). arXiv:2505.24050 [math.NT]
M. Ram Murty and V. Kumar Murty, A variant of the Hardy-Ramanujan theorem, Hardy-Ramanujan Journal, Vol. 44 (2021), pp. 32-40.
Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.

Even-indexed terms give A046886.
Cf. A000005, A001221, A001222, A002202, A027750, A064097, A141197, A185633, A202727, A202728, A322312, A322976, A333123, A346467, A355452.
Cf. A005408 (positions of 1's), A051222 (of 2's).

a067513 = sum . map (a010051 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Jul 31 2012

A067513 := proc(n)
    local a,d;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isprime(d+1) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A067513(n),n=1..100) ; # R. J. Mathar, Aug 07 2022

a[n_] := Length[Select[Divisors[n]+1, PrimeQ]]
Table[Count[Divisors[n],?(PrimeQ[#+1]&)],{n,110}] (* _Harvey P. Dale, Feb 29 2012 *)
a[n_] := DivisorSum[n, 1 &, PrimeQ[# + 1] &]; Array[a, 100] (* Amiram Eldar, Jan 11 2025 *)

a(n)=sumdiv(n,d,isprime(d+1)) \\ Charles R Greathouse IV, Dec 23 2011

from sympy import divisors, isprime
def a(n): return sum(1 for d in divisors(n, generator=True) if isprime(d+1))
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Jul 12 2022

a(n) = 2 iff Bernoulli number B_{n} has denominator 6 (cf. A051222). - Vladeta Jovovic, Feb 13 2002
a(n) <= A141197(n). - Reinhard Zumkeller, Oct 06 2008
a(n) = A001221(A027760(n)). - Enrique Pérez Herrero, Dec 23 2011
a(n) = Sum_{k = 1..A000005(n)} A010051(A027750(n,k)+1). - Reinhard Zumkeller, Jul 31 2012
a(n) = A001221(A185633(n)) = A001222(A322312(n)). - Antti Karttunen, Jul 12 2022
Sum_{k=1..n} a(k) = n * (log(log(n)) + B) + O(n/log(n)), where B is a constant (Prachar, 1955). - Amiram Eldar, Jan 11 2025

Edited by Dean Hickerson, Feb 12 2002

cp's OEIS Frontend

A045979 Bernoulli number B_{2n} 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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