A333635 - OEIS

A333635

Numbers m such that m^2 + 1 has at most 2 prime factors.

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

1,2

COMMENTS

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.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, A1.

LINKS

Table of n, a(n) for n=1..64.

Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Inventiones Mathematicae 47 (2) (1978), pp. 171-188.

EXAMPLE

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.

MATHEMATICA

Select[Range[100], PrimeQ[(k = #^2 + 1)] || PrimeOmega[k] == 2 &] (* Amiram Eldar, Mar 30 2020 *)