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Search: id:A107317
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| A107317 |
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Semiprimes of the form 2*(m^2+m+1) (implying that m^2+m+1 is a prime). |
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+0
1
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| 6, 14, 26, 62, 86, 146, 314, 422, 482, 614, 842, 926, 1202, 1514, 2246, 2966, 3446, 5102, 5942, 6614, 7082, 7814, 8846, 9662, 10226, 11402, 12014, 12326, 12962, 16022, 16382, 19802, 20606, 22262, 24422, 24866, 27614, 28562, 34586, 38366, 40046
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Also semiprimes n such that 2*n - 3 is a square. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Dec 29 2005. This coincidence was noticed by Andrew Plewe. Proof that this is the same sequence: If X is n^2+(n+1)^2+1, then 2X-3 is 4n^2+4n+1 = (2n+1)^2. And if 2X-3 is a square, then since it's odd, 2X-3 = (2n+1)^2 and X = n^2+(n+1)^2+1. - Don Reble, Apr 18 2007
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FORMULA
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2*A002383 = 2*(A002384^2+A002384+1).
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EXAMPLE
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a(1)=6 because 1^2 + 2^2 + 1 = 6 = 2*3;
a(2)=14 because 2^2 + 3^2 + 1 = 14 = 2*7;
a(3)=26 because 3^2 + 4^2 + 1 = 26 = 13*2.
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MATHEMATICA
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2(#^2 + # + 1) & /@ Select[ Range[144], PrimeQ[ #^2 + # + 1] &] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 28 2005)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2 && IntegerQ@Sqrt[2n - 3]; Select[ Range@43513, fQ[ # ] &] (* Robert G. Wilson v *)
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PROGRAM
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(PARI) for(n=2, 100000, if(bigomega(n)==2&&issquare(2*n-3), print1(n, ", "))) (Lambert Herrgesell)
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CROSSREFS
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Cf. A002383, A002384.
Adjacent sequences: A107314 A107315 A107316 this_sequence A107318 A107319 A107320
Sequence in context: A026055 A131951 A093776 this_sequence A071776 A063590 A128806
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KEYWORD
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easy,nonn
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AUTHOR
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Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), May 21 2005
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EXTENSIONS
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Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), May 28 2005
Re-edited by njas, Apr 18 2007
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