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

1,2

Comments

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

Links

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

Formula

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

Maple

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

Crossrefs

Cf. A000196, A295666, A296204.

Cf. A006881, A054753, A085986 (seem to be subsequences).

Cf. A004709, A007774, A056239, A112798, A118914, A307370, A325240.

Sequence in context: A299992 A237051 A340749 * A325281 A100658 A182301

Adjacent sequences: A296202 A296203 A296204 * A296206 A296207 A296208

Keyword

nonn

Author

Antti Karttunen, Dec 18 2017

Status

approved