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

Author

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

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

Magma

A185091 := func;[A185091(n):n in [3..94]];