A242776 Primes p such that 2^p + p^2 is not squarefree.

2, 11, 13, 29, 31, 47, 67, 83, 101, 103, 137, 139, 157, 173, 191, 193, 211, 227, 229, 263, 281, 283, 317, 337, 353, 373, 389, 397, 409, 421, 443, 461, 463, 479, 499, 569, 571, 587, 607, 641, 643, 659, 661, 677, 719, 733, 751, 769

Offset: 1

Author

Juri-Stepan Gerasimov, May 22 2014

Keywords

• nonn
• more

Comments

Sequence is infinite by Dirichlet's theorem: all primes which are 2 or 4 mod 9 are members. - Charles R Greathouse IV, May 27 2014

Additional terms: 823, 839, 857, 859, 877, 911, 929, 947, 953, 967, 983. - Kevin P. Thompson, Jun 13 2022

Examples

			2 is in this sequence because 2 is prime and 2^2 + 2^2 = 8 is divisible by 2^2.
11 is in this sequence because it is prime and 2^11 + 11^2 = 2169 is divisible by 3^2.
		

Links

• Kevin P. Thompson, Factorizations to support a(n) for n = 1..47

Crossrefs

A061238 and A061239 are subsequences.

Cf. A001580, A005117, A013929.

Programs

Magma

[n: n in [1..265] | IsPrime(n) and not IsSquarefree(2^n + n^2)];

Mathematica

Select[Prime[Range[25]], MoebiusMu[2^# + #^2] == 0 &] (* Alonso del Arte, May 26 2014 *)
Select[Range[100], PrimeQ[#] && ! SquareFreeQ[2^# + #^2] &] (* Amiram Eldar, Dec 24 2020 *)

PARI

s=[]; forprime(p=2, 300, if(!issquarefree(2^p+p^2), s=concat(s, p); print1(p, ", "))); s \\ Colin Barker, May 22 2014

Formula

a(n) < (4 + o(1))n log n. - Charles R Greathouse IV, May 27 2014

Extensions

a(21)-a(22) from Charles R Greathouse IV, May 26 2014

a(23)-a(30) from Charles R Greathouse IV, May 27 2014

a(31)-a(34) from Amiram Eldar, Dec 24 2020

a(35)-a(47) from Kevin P. Thompson, Jan 15 2022

a(48) from Kevin P. Thompson, Jun 13 2022