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A333635
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Numbers m such that m^2 + 1 has at most 2 prime factors.
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0
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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, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 58, 59, 60, 61, 62, 64, 65, 66, 69, 71, 74, 76, 78, 79, 80, 84, 85, 86, 88, 90, 92, 94, 95, 96, 100
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OFFSET
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1,2
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COMMENTS
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Equivalently, numbers m such that m^2 + 1 is prime or semiprime.
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).
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.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, A1.
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LINKS
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EXAMPLE
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10^2 + 1 = 101, which is prime, so 10 is in the sequence.
11^2 + 1 = 122 = 2 * 61, so 11 is in the sequence.
12^2 + 1 = 145 = 5 * 29, so 12 is in the sequence.
13^2 + 1 = 170 = 2 * 5 * 17, so 13 is not in the sequence.
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MATHEMATICA
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Select[Range[100], PrimeQ[(k = #^2 + 1)] || PrimeOmega[k] == 2 &] (* Amiram Eldar, Mar 30 2020 *)
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CROSSREFS
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Cf. A248742 (m^2 + 1 is prime or semiprime), this sequence (corresponding m).
Cf. A037143 (numbers with at most 2 prime factors counted with multiplicity).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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