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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

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: 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.)