A386991 -id:A386991 - cp's OEIS frontend

A387048 Numbers k such that k^2 + sopfr(k)^2 is prime, where sopfr = A001414.

6, 10, 12, 14, 21, 22, 39, 40, 44, 46, 51, 54, 57, 62, 65, 69, 74, 80, 82, 86, 90, 91, 95, 104, 108, 111, 115, 119, 129, 134, 141, 155, 161, 164, 166, 172, 176, 187, 189, 202, 210, 212, 217, 221, 226, 232, 244, 248, 252, 254, 265, 272, 274, 287, 292, 295, 297, 299, 300, 302, 305, 306, 328, 339

Offset: 1

Robert Israel, Aug 14 2025

nonn

Includes 2*p where p is a prime such that 5 * p^2 + 4 * p + 4 is prime. The Generalized Bunyakowsky Conjecture implies there are infinitely many of these.

			a(3) = 12 is a term because 12^2 + sopfr(12)^2 = 144 + (2*2+3)^2 = 193 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414, A050703, A386991, A387049.

sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
filter:= t -> isprime(t^2 + sopfr(t)^2):
select(filter, [$1..10^3]);

q[k_] := PrimeQ[k^2 + (Plus @@ Times @@@ FactorInteger[k])^2]; Select[Range[2, 340], q] (* Amiram Eldar, Aug 14 2025 *)

A385885 Nonprimes k such that k^2 - sopfr(k)^2 is a square, where sopfr = A001414.

Original entry on oeis.org

1, 4, 366, 1095, 51846, 258410, 982815, 10653351

Offset: 1

Robert Israel, Aug 13 2025

			a(4) = 1095 is a term because 1095 = 3 * 5 * 73 and 1095^2 - (3+5+73)^2 = 1192464 = 1092^2.

Cf. A001414, A386991.

sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
filter:= t -> not isprime(t) and issqr(t^2 - sopfr(t)^2):
select(filter, [$1..10^8]);

q[k_] := !PrimeQ[k] && IntegerQ[Sqrt[k^2 - (Plus @@ Times @@@ FactorInteger[k])^2]]; Select[Range[10^6], q] (* Amiram Eldar, Aug 14 2025 *)

isok(k) = if (!ispseudoprime(k), my(f=factor(k)); issquare(k^2 - (f[, 1]~*f[, 2])^2)); \\ Michel Marcus, Aug 21 2025

from math import isqrt
from sympy import factorint, isprime
def is_square(n): return n >= 0 and isqrt(n)**2 == n
def ok(n): return  not isprime(n) and is_square(n**2-sum(p*e for p,e in factorint(n).items())**2)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Aug 14 2025

Edited to insert 1 by Robert Israel, Aug 22 2025

cp's OEIS Frontend

A387048 Numbers k such that k^2 + sopfr(k)^2 is prime, where sopfr = A001414.

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