A296205 - OEIS

%I #25 Feb 20 2024 14:44:21

%S 1,6,10,12,14,15,18,20,21,22,26,28,33,34,35,36,38,39,44,45,46,50,51,

%T 52,55,57,58,62,63,65,68,69,74,75,76,77,82,85,86,87,91,92,93,94,95,98,

%U 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

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

%C 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

%H Robert Israel, <a href="/A296205/b296205.txt">Table of n, a(n) for n = 1..10000</a>

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

%p filter:= proc(k) local d,r,v;

%p    r:= 1;

%p    for d in numtheory:-divisors(k^2) do

%p      v:= igcd(d,k^2/d);

%p      if isprime(v) then r:= r*v fi

%p    od;

%p    r = k^2

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, Feb 20 2024

%Y Cf. A000196, A295666, A296204.

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

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

%K nonn

%O 1,2

%A _Antti Karttunen_, Dec 18 2017