A396455 Primes p such that p^2 = A075254(k) for some k.

2, 11, 31, 43, 47, 67, 71, 73, 103, 107, 131, 137, 149, 167, 193, 197, 199, 211, 223, 251, 263, 271, 277, 281, 283, 311, 313, 317, 337, 347, 353, 359, 367, 373, 379, 383, 401, 433, 439, 443, 457, 463, 467, 479, 487, 491, 509, 521, 557, 563, 571, 599, 601, 607, 613, 617, 619, 641, 643, 659, 661

Offset

1,1

Comments

Primes p such that k + sopfr(k) = p^2 has at least one solution, where sopfr(k) = A001414(k) is the sum of prime factors of k with multiplicity.

Links

Robert Israel, Table of n, a(n) for n = 1..1065

Example

a(3) = 31 is a term because 31 is a prime and 31^2 = 961 = A075254(936).
The first term p for which k + sopfr(k) = p^2 has more than one solution is a(8) = 73, as 73^2 = 5329 = A075254(5008) = A075254(5278).
The first term for which there are 3 solutions is a(9) = 103, as 103^2 = 10609 = A075254(10486) = A075254(10545) = A075254(10556).

Maple

N:= 2*10^8: # for terms <= sqrt(N)
P:= {}:
for x from 1 to N do
   v:= x + add(s[1]*s[2], s=ifactors(x)[2]);
   if v <= N and issqr(v) then
     p:= sqrt(v); if isprime(p) then P:= P union {p} fi;
   fi;
od:
sort(convert(P, list));

Crossrefs

Cf. A001414, A075254, A396408.

Sequence in context: A187830 A115058 A158295 * A213898 A085041 A197642

Adjacent sequences: A396452 A396453 A396454 * A396456 A396457 A396458

Keyword

nonn

Author

Will Gosnell and Robert Israel, May 26 2026

Status

approved