A185091 The smallest positive noncomposite q such that 2n-1 = 2p+q for some positive noncomposite p.

1, 1, 1, 3, 1, 3, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 17, 11, 1, 3, 1, 3, 5, 7, 13, 11, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 17, 11, 1, 3, 5, 7, 29, 11, 1, 3, 1, 3, 5, 7, 13, 11, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 13, 11, 1, 3, 5, 7, 1, 3, 5, 7, 17, 11, 1, 3, 5, 7, 29

Offset

2,4

Comments

It is a Goldbach conjecture variant that terms exist for 2n-1 >= 5.

Lemma: N=2n-1 is coprime to q=a(n) unless N=3q. Proof: Suppose N and q are not coprime; so we have N=2p+q=iq with i=/=1=/=q, so (i-1)q=2p; now since q=/=2 (because N is odd), then q=p and i=3. QED.

Empirically, N=3q only for N=9,21.

References

Emile Lemoine, L'intermédiaire des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.

Links

Jason Kimberley, Table of n, a(n) for n = 2..10002 (corrected by Michel Marcus, Jan 19 2019)

Brian H. Mayoh, On the second Goldbach conjecture, BIT Numerical Mathematics 6 (1966) 1, 48-50

Wikipedia, Lemoine's conjecture

Index entries for sequences related to Goldbach conjecture

Prog

(Magma) A185091 := func<n|exists(q){q:q in[1..N div 3 by 2]|(q eq 1 or IsPrime(q))and IsPrime((N-q)div 2)}select q else -1 where N is 2*n-1>; [A185091(n):n in [3..94]];

Crossrefs

Records in this sequence are in A002092 occurring at 2n-1 in A002091.

Sequence in context: A372832 A176246 A046933 * A332031 A023511 A035628

Adjacent sequences: A185088 A185089 A185090 * A185092 A185093 A185094

Keyword

nonn,easy

Author

Jason Kimberley (with thanks to Hugo Pfoertner), Sep 05 2011

Status

approved