A079148 -id:A079148 - cp's OEIS frontend

A119660 Prime factor of the distinct numbers appearing as denominators of Bernoulli numbers A090801 that is greater than all previous a(n). a(1) = 2.

2, 3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 239, 263, 347, 359, 383, 443, 467, 479, 503, 563, 587, 647, 659, 719, 827, 839, 863, 887, 983, 1019, 1187, 1223, 1259, 1283, 1307, 1319, 1367, 1439, 1487, 1499, 1523, 1619, 1787

Offset: 1

Alexander Adamchuk, Jul 28 2006

nonn

a(n) is identical to A079148[n] up to a(14)=227. Most a(n) except 2,3,239,443,647,659,827,1223,1259,1499,1787... belong to A005385[n]: Safe primes p: (p-1)/2 is also prime.

Except for 2 and 3, the same as A092307. - T. D. Noe, Sep 25 2006

			A090801[n] begins {1, 2, 6, 30, 42, 66, 138, 282, 330, 354, 498, 510, 642, 690, ...} = {1, {2,1}, {2,3}, {2,3,5}, {2,3,7}, {2,3,11}, {2,3,23}, {2,3,47}, {2,3,5,11}, {2,3,59}, {2,3,83}, {2,3,5,17}, {2,3,107}, {2,3,5,23}, ...}.
a(1) = 2, a(2) = 3, a(3) = 5, a(4) = 7, a(5) = 11, a(6) = 23, a(7) = 47, a(8) = 59, a(9) = 83, a(10) = 107.

Cf. A090801, A079148, A005385.

A303705 a(1) = 3; a(n) is the smallest prime such that gcd(a(i)-1, a(n)-1) = 2 holds for 1 <= i < n.

Original entry on oeis.org

3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 239, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1223, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2243, 2447

Offset: 1

Jianing Song, Apr 29 2018

nonn

a(n) exists for all n, which is easily shown by Dirichlet's theorem on arithmetic progressions.
Apart from 3, the first term that is not a term in A005385 is 239. The first term in A092307 and A119660 (apart from 2) that is not a term here is 443.
Clearly all safe primes are in this sequence, and all terms except a(2) = 5 are == 3 (mod 4).

			a(13) = 239 since lcm(a(1)-1, a(2)-1, ..., a(12)-1) = 2^2*3*5*11*23*29*41*53*83*89*113 and 239-1 = 2*7*17.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005385, A079148, A092307, A119660.

A[1]:= 3: L:= 2:
for i from 2 to 100 do
  p:= nextprime(A[i-1]);
  while igcd(L, p-1) > 2 do p:= nextprime(p) od:
  A[i]:= p;
  L:= ilcm(L, p-1);
od:
seq(A[i],i=1..100); # Robert Israel, Apr 29 2018

Corrected by Robert Israel, Apr 29 2018

A119653 Denominator of BernoulliB[2p] divided by 6, where p=Prime[n].

Original entry on oeis.org

5, 7, 11, 1, 23, 1, 1, 1, 47, 59, 1, 1, 83, 1, 1, 107, 1, 1, 1, 1, 1, 1, 167, 179, 1, 1, 1, 1, 1, 227, 1, 263, 1, 1, 1, 1, 1, 1, 1, 347, 359, 1, 383, 1, 1, 1, 1, 1, 1, 1, 467, 479, 1, 503, 1, 1, 1, 1, 1, 563, 1, 587, 1, 1, 1, 1, 1, 1, 1, 1, 1, 719, 1, 1, 1, 1, 1, 1, 1, 1, 839, 1, 863, 1, 1

Offset: 1

Alexander Adamchuk, Jul 28 2006

nonn

a(n) is equal to 1 or a safe prime p: (p-1)/2 is also prime, A005385[n] = 5,7,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,587,719,839,863,887,983,1019... The indices of primes in a(n) are n=1,2,3,5,9,10,13,16,23,24..=A072192[n] Indices of Sophie Germain primes: p and 2p+1 are primes.

Cf. A002445, A005385, A072192, A005384, A079148.

Table[Denominator[BernoulliB[2Prime[n]]]/6,{n,1,100}]

a(n) = Denominator[BernoulliB[2Prime[n]]]/6.

cp's OEIS Frontend

A119660 Prime factor of the distinct numbers appearing as denominators of Bernoulli numbers A090801 that is greater than all previous a(n). a(1) = 2.

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A303705 a(1) = 3; a(n) is the smallest prime such that gcd(a(i)-1, a(n)-1) = 2 holds for 1 <= i < n.

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A119653 Denominator of BernoulliB[2p] divided by 6, where p=Prime[n].

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