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

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

Offset

1,2

Comments

a(9) > 10^8. - Tyler Busby, Sep 06 2025

a(9) > 2 * 10^8. - Tyler Yu, Feb 02 2026

Links

Table of n, a(n) for n=1..8.

Example

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

Maple

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]);

Mathematica

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

Prog

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

Crossrefs

Cf. A001414, A386991.

Sequence in context: A109760 A051181 A326996 * A154682 A292195 A154569

Adjacent sequences: A385882 A385883 A385884 * A385886 A385887 A385888

Keyword

nonn,more

Author

Robert Israel, Aug 13 2025

Extensions

Edited to insert 1 by Robert Israel, Aug 22 2025

Status

approved