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

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

Author

Robert G. Wilson v, Aug 19 2010

Keywords

• nonn

Comments

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)

Links

• 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]

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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