A361300 Numbers of the form m^2 + p^2 for p prime and m > 0.

5, 8, 10, 13, 18, 20, 25, 26, 29, 34, 40, 41, 45, 50, 53, 58, 61, 65, 68, 73, 74, 85, 89, 90, 98, 104, 106, 109, 113, 122, 125, 130, 137, 146, 148, 149, 153, 157, 169, 170, 173, 178, 185, 193, 194, 200, 202, 205, 218, 221, 229, 233, 234, 242

Offset: 1

Author

Charles R Greathouse IV, Mar 08 2023

Keywords

• nonn
• easy

Comments

Rieger proves that there are >> x/log x terms of this sequence up to x, and together with the trivial upper bound << x/log x this shows that a(n) ≍ n log n. (Rieger does not prove that a(n) ~ n log n, the constant factor may be larger.)

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• G. J. Rieger, Über die Summe aus einem Quadrat und einem Primzahlquadrat, J. Reine Angew. Math. 231 (1968), 89-100.

Crossrefs

Subsequence of A000404; A185086 is a subsequence.

Programs

PARI

list(lim)=my(v=List()); forprime(p=2,sqrtint(lim\=1), my(p2=p^2); for(m=1,sqrtint(lim-p2), listput(v,p2+m^2))); Set(v)