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%I #38 Sep 08 2022 08:45:38
%S 10,26,65,82,122,145,226,362,485,626,785,842,901,1157,1226,1522,1765,
%T 1937,2026,2117,2305,2402,2501,2602,2705,3365,3482,3601,3722,3845,
%U 4097,4226,4762,5042,5777,6085,6242,6401,7226,7397,7745,8465,9026,9217,10001,10202
%N Semiprimes of the form k^2+1.
%C 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.
%C 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
%H Charles R Greathouse IV, <a href="/A144255/b144255.txt">Table of n, a(n) for n = 1..10000</a>
%H Henryk Iwaniec, <a href="http://dx.doi.org/10.1007/BF01578070">Almost-primes represented by quadratic polynomials</a>, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.
%F a(n) = A085722(n)^2 + 1.
%F Equals { n^2+1 | A193432(n)=2 }. - _M. F. Hasler_, Mar 11 2012
%t Select[Table[n^2 + 1, {n, 100}], PrimeOmega[#] == 2&] (* _Vincenzo Librandi_, Sep 22 2012 *)
%o (PARI) select(n->bigomega(n)==2,vector(500,n,n^2+1)) \\ Zak Seidov Feb 24 2011
%o (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
%o (Python)
%o from sympy import primeomega
%o from itertools import count, takewhile
%o def aupto(limit):
%o form = takewhile(lambda x: x <= limit, (k**2+1 for k in count(1)))
%o return [number for number in form if primeomega(number)==2]
%o print(aupto(10202)) # _Michael S. Branicky_, Oct 26 2021
%Y Cf. A001358, A085722, A069987, A193432.
%Y Subsequence of A134406.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, Sep 16 2008
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