A381330 Numbers that are the sum of a prime and the square of a prime in more than one way.

11, 27, 28, 32, 38, 51, 52, 54, 56, 62, 66, 68, 72, 78, 80, 86, 92, 96, 98, 108, 110, 116, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 156, 158, 162, 164, 171, 172, 174, 176, 180, 182, 186, 188, 192, 198, 200, 204, 206, 210, 212, 216, 218, 222, 224, 228

Offset: 1

Author

Chai Wah Wu, Feb 20 2025

Keywords

• nonn

Comments

Subsequence of A081053. Most terms are even. The odd terms are 11, 27, 51, 171, 363, 843, 1371, 1851 and must be of the form 2+p^2=4+q for primes p, q. In particular, the odd terms are exactly A049002(n)+4 for n>1.

Examples

			11 is a term since 11 = 2^2+7  = 3^2+2.
27 is a term since 27 = 2^2+23 = 5^2+2.
28 is a term since 28 = 3^2+19 = 5^2+3.
32 is a term since 32 = 3^2+23 = 5^2+7.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A081053, A049002.

Programs

PARI

isok(k) = my(nb=0); forprime(p=2, sqrtint(k), if (isprime(k-p^2), nb++);); nb > 1; \\ Michel Marcus, Feb 21 2025

Python

from math import isqrt
from itertools import count, islice
from sympy import isprime, primerange
def A381330_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = 0
        for p in primerange(isqrt(n)+1):
            if isprime(n-p**2):
                c += 1
            if c>1:
                yield n
                break
A381330_list = list(islice(A381330_gen(),30))