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A069482
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Prime(n+1)^2 - prime(n)^2.
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13
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5, 16, 24, 72, 48, 120, 72, 168, 312, 120, 408, 312, 168, 360, 600, 672, 240, 768, 552, 288, 912, 648, 1032, 1488, 792, 408, 840, 432, 888, 3360, 1032, 1608, 552, 2880, 600, 1848, 1920, 1320, 2040, 2112, 720, 3720, 768, 1560, 792, 4920, 5208, 1800, 912, 1848
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OFFSET
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1,1
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COMMENTS
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a(n)=A001248(n+1)-A001248(n)=A000040(n+1)^2-A000040(n)^2=
=(A000040(n+1)-A000040(n))*(A000040(n+1)+A000040(n))=
=A001223(n)*A001043(n);
together with A069484(n) and A069486(n) a Pythagorean triangle is formed with area = A069487(n).
For n>2: A078701(a(n)) = 3.
Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), May 28 2009: (Start)
Except for the first two terms, these numbers are divisible by 24. Let
p,q be consecutive primes. Then p > 3 = 3k+-1 and q = 3m+-1 and
(3k+-1)^2 - (3m+-1)^2 is divisible by 3. Similarly, p = 4k+-1 and q=4m+-1 and
(4k+-1)^2 - (4m+-1)^2 is divisible by 8. So 8 and 3 divide q^2-p^2 => 24
divides q^2-p^2. (End)
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LINKS
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R. Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pythagorean Triple
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EXAMPLE
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A000040(10)=29, A000040(10+1)=31, A001248(10)=841, A001248(10+1)=961, a(10)=961-841=120, A069486(10)=2*31*29=1798, A069484(10)=961+841=1802:
120^2 + 1798^ = 14400 + 3232804 = 3247204 = 1802^2.
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MATHEMATICA
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a=4; lst={}; Do[p=Prime[n]; AppendTo[lst, p^2-a]; a=p^2, {n, 2, 6!}]; lst [From Vladimir Joseph Stephan Orlovsky, Mar 01 2009]
#[[2]]-#[[1]]&/@Partition[Prime[Range[60]]^2, 2, 1] [From Harvey P. Dale, Jan. 13, 2011]
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CROSSREFS
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Cf. A069483.
Sequence in context: A063243 A087747 A090785 * A102045 A055508 A171787
Adjacent sequences: A069479 A069480 A069481 * A069483 A069484 A069485
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller, Mar 29 2002, Aug 05 2007
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STATUS
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approved
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