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Search: id:A075555
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| A075555 |
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Smallest prime p such that p+n is a square, or 0 if no such p exists. |
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+0
4
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| 3, 2, 13, 5, 11, 3, 2, 17, 7, 71, 5, 13, 3, 2, 181, 0, 19, 7, 17, 5, 43, 3, 2, 97, 11, 23, 37, 53, 7, 19, 5, 17, 3, 2, 29, 13, 107, 11, 61, 41, 23, 7, 101, 5, 19, 3, 2, 73, 0, 31, 13, 29, 11, 67, 89, 113, 7, 23, 5, 61, 3, 2, 37, 17, 79, 103, 257, 13, 31, 11, 29, 97, 71, 7, 181, 5
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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If n=A047845(i)^2 for some i, i.e. if n has the form ((k-1)/2)^2 with k odd but not prime, then a(n)=0. It is conjectured that these are the only values of n for which a(n)=0; this would follow from Schinzel's hypothesis.
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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EXAMPLE
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a(8) = 17 because 8 + 17 is the first square that can be made by adding a prime to 8.
a(16) = 0 because 16 + p cannot be x^2, since then p = x^2 - 16 = (x-4)(x+4).
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MATHEMATICA
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a[n_] := If[IntegerQ[s=Sqrt[n]]&&!PrimeQ[2s+1], 0, For[x=Ceiling[s], True, x++, If[PrimeQ[x^2-n], Return[x^2-n]]]]
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PROGRAM
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(PARI) for(n=1, 100, f=0:forprime(p=2, 10^7, if(issquare(p+n), f=p:break)):if(f, print1(f", "), print1("0, ")))
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CROSSREFS
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Cf. A075556.
a(n) = A105016(n)^2 - n, if a(n) exists.
Sequence in context: A125135 A055456 A093922 this_sequence A075556 A087357 A131050
Adjacent sequences: A075552 A075553 A075554 this_sequence A075556 A075557 A075558
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 23 2002
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EXTENSIONS
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More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 28 2003
Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Mar 31 2003
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