A232009 -id:A232009 - cp's OEIS frontend

A067775 Primes p such that p + 4 is composite.

2, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 359, 367, 373, 383, 389

Offset: 1

Benoit Cloitre, Feb 06 2002

nonn

Primes n such that n!*B(n+3) is an integer where B(k) are the Bernoulli numbers B(1) = -1/2, B(2) = 1/6, B(4) = -1/30, ..., B(2m+1) = 0 for m > 1.
If n is prime n!*B(n-1) is always an integer. Note that if Goldbach's conjecture (2n = p1 + p2 for all n >= 2) is false and K is the smallest value of n for which it fails, then for 2(K-2) = p3 + p4, the primes p3 and p4 must be taken from this list. See similar comment for A140555. - Keith Backman, Apr 06 2012
Complement of A023200 (primes p such that p + 4 is also prime) with respect to A000040 (primes). For p > 2: primes p such that there is no prime of the form r^2 + p where r is prime, subsequence of A232010. Example: the prime 7 is not in the sequence because 2^2 + 7 = 11 (prime). A232009(a(n)) = 0 for n > 1 . - Jaroslav Krizek, Nov 22 2013

Klaus Brockhaus, Table of n, a(n) for n = 1..1026
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A049591, A140555, A232009, A232010, A232012, A023200.

A067775 = {}; Do[p = Prime@ n; If[ IntegerQ[ p! BernoulliB[p + 3]], AppendTo[A067775, p]], {n, 77}]; A067775 (* Robert G. Wilson v, Aug 19 2008 *)
Select[Prime[Range[80]], Not[PrimeQ[# + 4]] &] (* Alonso del Arte, Apr 02 2014 *)

lista(nn) = {forprime(p=1, nn, if (! isprime(p+4), print1(p, ", ")););} \\ Michel Marcus, Nov 22 2013

a(n) ~ n log n. - Charles R Greathouse IV, Nov 22 2013

New name from Klaus Brockhaus at the suggestion of Michel Marcus, Nov 22 2013

A232012 Numbers n such that p^2 + n is prime for some prime p.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 22, 24, 25, 27, 28, 30, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 46, 48, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 64, 66, 67, 69, 70, 72, 74, 75, 76, 78, 79, 80, 82, 84, 85, 88, 90, 92, 93, 94, 96, 97, 98, 99, 100

Offset: 1

Jaroslav Krizek, Nov 22 2013

nonn

Supersequence of A023200 (primes p such that p + 4 is also prime). Complement of A232010.

			2 is in sequence because 3^2 + 2 = 11 (3 and 11 are primes).

Cf. A232009, A232010, A232011, A023200.

Module[{nn=50,prs},prs=Prime[Range[nn]]^2;Select[Range[ 100],AnyTrue[ prs+#,PrimeQ]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 04 2020 *)

A232010 Numbers k such that p^2 + k is composite for all prime p.

Original entry on oeis.org

5, 11, 17, 21, 23, 26, 29, 31, 35, 41, 45, 47, 51, 53, 56, 59, 61, 65, 68, 71, 73, 77, 81, 83, 86, 87, 89, 91, 95, 101, 107, 110, 111, 113, 115, 116, 117, 119, 121, 125, 129, 131, 134, 137, 139, 141, 143, 146, 149, 151, 152, 155, 157, 161, 165, 167, 171, 173, 176

Offset: 1

Jaroslav Krizek, Nov 20 2013

nonn

Primes from this sequence are in A067775.

Complement of A232012.

			7 is not in the sequence because 2^2 + 7 = 11 (prime).

Cf. A232009, A067775, A232012, A023200.

A232009(a(n)) = 0.

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A067775 Primes p such that p + 4 is composite.

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A232012 Numbers n such that p^2 + n is prime for some prime p.

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A232010 Numbers k such that p^2 + k is composite for all prime p.

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