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 A185091 The smallest positive noncomposite q such that 2n-1 = 2p+q for some positive noncomposite p. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 PROG (MAGMA) A185091 := func; [A185091(n):n in [3..94]]; CROSSREFS Records in this sequence are in A002092 occurring at 2n-1 in A002091. Sequence in context: A066839 A176246 A046933 * A023511 A035628 A187562 Adjacent sequences:  A185088 A185089 A185090 * A185092 A185093 A185094 KEYWORD nonn,easy AUTHOR Jason Kimberley (with thanks to Hugo Pfoertner), Sep 05 2011 STATUS approved

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Last modified August 18 04:50 EDT 2019. Contains 326072 sequences. (Running on oeis4.)