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

2,1

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

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]

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A010051, A002378, A000217.

Sequence in context: A121795 A253027 A249384 * A364564 A188889 A219604

Adjacent sequences: A228443 A228444 A228445 * A228447 A228448 A228449

Keyword

easy,nonn

Author

Bill McEachen, Oct 26 2013

Extensions

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

Status

approved