This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333635 #48 Aug 08 2025 19:45:42 %S A333635 1,2,3,4,5,6,8,9,10,11,12,14,15,16,19,20,22,24,25,26,28,29,30,34,35, %T A333635 36,39,40,42,44,45,46,48,49,50,51,52,54,56,58,59,60,61,62,64,65,66,69, %U A333635 71,74,76,78,79,80,84,85,86,88,90,92,94,95,96,100 %N A333635 Numbers m such that m^2 + 1 has at most 2 prime factors. %C A333635 Equivalently, numbers m such that m^2 + 1 is prime or semiprime. %C A333635 Henryk Iwaniec proved in 1978 that this sequence is infinite (see link). By contrast, it is not known whether there are infinitely many primes of the form m^2 + 1 (or infinitely many semiprimes of that form). %C A333635 The integers that have at most 2 prime factors counted with multiplicity are called almost-primes of order 2 and they are in A037143. Here, as m^2 + 1 is not a square for m > 0, all the semiprimes of this form have two distinct prime factors (A144255), and with the primes of the form m^2 + 1 (A002496), they constitute A248742. %D A333635 R. K. Guy, Unsolved Problems in Number Theory, A1. %H A333635 Harvey P. Dale, <a href="/A333635/b333635.txt">Table of n, a(n) for n = 1..1000</a> %H A333635 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. %e A333635 10^2 + 1 = 101, which is prime, so 10 is in the sequence. %e A333635 11^2 + 1 = 122 = 2 * 61, so 11 is in the sequence. %e A333635 12^2 + 1 = 145 = 5 * 29, so 12 is in the sequence. %e A333635 13^2 + 1 = 170 = 2 * 5 * 17, so 13 is not in the sequence. %t A333635 Select[Range[100], PrimeQ[(k = #^2 + 1)] || PrimeOmega[k] == 2 &] (* _Amiram Eldar_, Mar 30 2020 *) %t A333635 Select[Range[100],PrimeOmega[#^2+1]<3&] (* _Harvey P. Dale_, Aug 08 2025 *) %Y A333635 Union of A005574 and A085722. %Y A333635 Cf. A002496 (m^2 + 1 is prime), A005574 (corresponding m). %Y A333635 Cf. A144255 (m^2 + 1 is semiprime), A085722 (corresponding m). %Y A333635 Cf. A248742 (m^2 + 1 is prime or semiprime), this sequence (corresponding m). %Y A333635 Cf. A037143 (numbers with at most 2 prime factors counted with multiplicity). %K A333635 nonn %O A333635 1,2 %A A333635 _Bernard Schott_, Mar 30 2020