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A228446
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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.
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4
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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
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OFFSET
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2,1
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COMMENTS
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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.
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REFERENCES
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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.)
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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]
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Nov 11 2020 (including addition of escape clause).
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STATUS
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approved
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