A296204 -id:A296204 - cp's OEIS frontend

A296205 Numbers k such that Product_{d|k^2, gcd(d,k^2/d) is prime} gcd(d,k^2/d) = k^2.

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 36, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 141, 142, 143, 145, 146, 147, 148, 153, 155, 158, 159, 161

Offset: 1

Antti Karttunen, Dec 18 2017

nonn

Except for a(1) = 1, these appear to be cubefree numbers with two distinct prime factors, or Heinz numbers of integer partitions with two distinct parts, none appearing more than twice. The enumeration of these partitions by sum is given by A307370. Equivalently, except for a(1) = 1, this sequence is the intersection of A004709 and A007774. - Gus Wiseman, Jul 03 2019

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

Cf. A000196, A295666, A296204.
Cf. A006881, A054753, A085986 (seem to be subsequences).
Cf. A004709, A007774, A056239, A112798, A118914, A307370, A325240.

filter:= proc(k) local d,r,v;
   r:= 1;
   for d in numtheory:-divisors(k^2) do
     v:= igcd(d,k^2/d);
     if isprime(v) then r:= r*v fi
   od;
   r = k^2
end proc:
select(filter, [$1..200]); # Robert Israel, Feb 20 2024

a(n) = A000196(A296204(n)).

cp's OEIS Frontend

A296205 Numbers k such that Product_{d|k^2, gcd(d,k^2/d) is prime} gcd(d,k^2/d) = k^2.

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