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A228446 For odd n >= 5, lowest prime p such that n = p + x*(x+1) for some x > 0. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

This is Sun's conjecture 1.4 in the paper listed 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.  An example is 19 encountered from n=21=3*7, 19>7.

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.

51 = 31+4*5  where no lower odd prime provides a solution for odd 51.

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 * A188889 A219604 A253398

Adjacent sequences:  A228443 A228444 A228445 * A228447 A228448 A228449

KEYWORD

easy,nonn

AUTHOR

Bill McEachen, Oct 26 2013

STATUS

approved

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Last modified August 18 22:09 EDT 2017. Contains 290768 sequences.