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Search: id:A103828
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| A103828 |
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Sequence of odd numbers defined recursively by: a(1)=1 and a(n) is the first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for 1<=i<=n-1. |
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+0
8
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| 1, 3, 9, 27, 69, 429, 1059, 56499, 166839, 5020059, 7681809, 274343589, 8316187179
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Is the sequence infinite? Is each prime a(i)+a(j)+1, i<>j, always distinct?
Except for a(1), a(n) == 3 (mod 6). - Robert G. Wilson v Jun 02 2006.
The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n>2, a(n)+2 and a(n)+4 are both primes, so a proof that this sequence is infinite would also show that there are infinitely many twin primes. - N. J. A. Sloane (njas(AT)research.att.com), Apr 21 2007
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REFERENCES
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G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio Numerorum' III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
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EXAMPLE
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a(1)=1, a(2)=3, but 5+1+1=7, 5+3+1=9; 7+1+1=9, 7+3+1=11; 9+1+1=11, 9+3+1=13 so a(3)=9.
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MAPLE
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EP:=[]: for w to 1 do for n from 1 to 8*10^6 do s:=2*n-1; Q:=map(z->z+s+1, EP); if andmap(isprime, Q) then EP:=[op(EP), s]; print(nops(EP), s); fi od od; EP;
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MATHEMATICA
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a[1] = 1; a[2] = 3; a[n_] := a[n] = Block[{k = a[n - 1] + 6, t = Table[ a[i], {i, n - 1}] + 1}, While[ First@ Union@ PrimeQ[k + t] == False, k += 6]; k]; Do[ Print[ a[n]], {n, 15}] - Robert G. Wilson v (rgwv(at)rgwv.com), Jun 03 2006
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CROSSREFS
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Cf. A118818, A093483, A128933 (a(n)+1), A115760 (2*a(n)+1). Primes arising from this sequence are in A115782.
Sequence in context: A161712 A137368 A036215 this_sequence A110740 A042938 A084707
Adjacent sequences: A103825 A103826 A103827 this_sequence A103829 A103830 A103831
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KEYWORD
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easy,nonn
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AUTHOR
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Walter Kehowski (wkehowski(AT)cox.net), May 29 2006
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EXTENSIONS
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a(12) from Robert G. Wilson v (rgwv(at)rgwv.com), Jun 03 2006
a(13) from Walter Kehowski (wkehowski(AT)cox.net), Jun 03 2006
Corrected definition. - walter kehowski (wkehowski(AT)cox.net), Nov 03 2008
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