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A144255
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Semiprimes of the form k^2+1.
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21
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10, 26, 65, 82, 122, 145, 226, 362, 485, 626, 785, 842, 901, 1157, 1226, 1522, 1765, 1937, 2026, 2117, 2305, 2402, 2501, 2602, 2705, 3365, 3482, 3601, 3722, 3845, 4097, 4226, 4762, 5042, 5777, 6085, 6242, 6401, 7226, 7397, 7745, 8465, 9026, 9217, 10001, 10202
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OFFSET
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1,1
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COMMENTS
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Iwaniec proves that there are an infinite number of semiprimes or primes of the form n^2+1. Because n^2+1 is not a square for n>0, all such semiprimes have two distinct prime factors.
Moreover, this implies that one prime factor p of n^2+1 is strictly smaller than n, and therefore also divisor of (the usually much smaller) m^2+1, where m = n % p (binary "mod" operation). - M. F. Hasler, Mar 11 2012
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LINKS
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FORMULA
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MATHEMATICA
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Select[Table[n^2 + 1, {n, 100}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)
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PROG
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(PARI) select(n->bigomega(n)==2, vector(500, n, n^2+1)) \\ Zak Seidov Feb 24 2011
(Magma) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..100] | IsSemiprime(s) where s is n^2 + 1]; // Vincenzo Librandi, Sep 22 2012
(Python)
from sympy import primeomega
from itertools import count, takewhile
def aupto(limit):
form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
return [number for number in form if primeomega(number)==2]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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