A228446 a(n) = smallest prime p such that 2*n+1 = p + x*(x+1) for some positive integer x, or -1 if no such prime exists.

3, 5, 3, 5, 7, 3, 5, 7, 19, 3, 5, 7, 17, 11, 3, 5, 7, 19, 11, 13, 3, 5, 7, 31, 11, 13, 37, 3, 5, 7, 23, 11, 13, 29, 17, 3, 5, 7, 61, 11, 13, 31, 17, 19, 3, 5, 7, 43, 11, 13, 103, 17, 19, 109, 3, 5, 7, 29, 11, 13, 53, 17, 19, 41, 23, 3, 5, 7, 31, 11, 13, 37

Offset: 2

Author

Bill McEachen, Oct 26 2013

Keywords

• easy
• nonn

Comments

Based on Sun's conjecture 1.4 in the paper referenced below.

The plot shows an ever-widening band of sawtooth shape. New maxima values will include sequence members larger than the largest prime factor of the original n. For example when n = 21 with prime factors 3 and 7, and a(10) = 19.

a(A000124(n)) = 3; a(A133263(n)) = 5; a(A167614(n)) = 7. - Reinhard Zumkeller, Mar 12 2014

Examples

			21 = 19+1*2 where no solution exists using p = 2, 3, 5, 7, 11, 13, 17. So a(10) = 19.
51 = 31+4*5 where no lower odd prime provides a solution. So a(25) = 31.
		

References

• Z. W. Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory 1(2009), no. 1, 65-76. (See Conjecture 1.4.)

Links

• T. D. Noe, Table of n, a(n) for n = 2..1000
• Z. W. Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.
• Christian Krause, et al, A mined LODA assembly source for this sequence (conjectured)

Crossrefs

Cf. A010051, A002378, A000217.

Programs

Haskell

a228446 n = head
   [q | let m = 2 * n + 1,
        q <- map (m -) $ reverse $ takeWhile (< m) $ tail a002378_list,
        a010051 q == 1]
-- Reinhard Zumkeller, Mar 12 2014

Mathematica

nn = 14; ob = Table[n*(n+1), {n, nn}]; Table[p = Min[Select[n - ob, # > 0 && PrimeQ[#] &]]; p, {n, 5, ob[[-1]], 2}] (* T. D. Noe, Oct 27 2013 *)

PARI

a(n) = {oddn = 2*n+1; x = oddn; while (! isprime(oddn - x*(x+1)), x--); oddn - x*(x+1);} \\ Michel Marcus, Oct 27 2013

Extensions

Entry revised by N. J. A. Sloane, Nov 11 2020 (including addition of escape clause).