A284018 -id:A284018 - cp's OEIS frontend

A038109 Divisible exactly by the square of a prime.

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242

Offset: 1

Felice Russo

Numbers for which at least one prime factor exponent is exactly 2.
Sometimes called squarefull numbers, although that term is usually reserved for A001694. - N. J. A. Sloane, Jul 22 2012
The asymptotic density of this sequence is 1 - A330596 = 0.2514647... - Amiram Eldar, Aug 12 2020

			20=5*2*2 is divisible by 2^2.

R. J. Mathar, Table of n, a(n) for n = 1..1247

Cf. A001694, A013929, A284017, A284018.

isA038109 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) = 2 then
            return true;
        end if;
    end do:
    false ;
end proc: # R. J. Mathar, Dec 08 2015
# second Maple program:
q:= n-> ormap(i-> i[2]=2, ifactors(n)[2]):
select(q, [$1..300])[];  # Alois P. Heinz, Aug 12 2020

Select[Range[250],MemberQ[Transpose[FactorInteger[#]][[2]],2]&] (* Harvey P. Dale, Sep 24 2012 *)

is(n)=#select(n->n==2, Set(factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

Corrected and extended by Erich Friedman

A284017 Square root of the smallest square referenced in A038109 (Divisible exactly by the square of a prime).

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 2, 2, 3, 7, 5, 2, 2, 3, 2, 3, 5, 2, 2, 3, 2, 7, 3, 2, 2, 2, 3, 11, 2, 3, 2, 2, 3, 7, 2, 5, 3, 2, 2, 13, 3, 2, 5, 2, 2, 2, 3, 5, 2, 3, 2, 2, 3, 2, 3, 2, 11, 2, 7, 2, 2, 3, 2, 5, 2, 3, 2, 3, 17, 2, 7, 2, 3, 2, 3, 2, 2, 5, 2, 3, 13, 2, 3, 2

Offset: 1

Robert Price, Mar 18 2017

nonn

a(n) is the least prime p whose exponent in the prime factorization of A038109(n) is exactly 2. - Robert Israel, Mar 28 2017

			A038109(3)=12, 12 = 2*2*3, so 12 is divisible by the square of 2.

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

Cf. A038109, A284018.

N:= 1000: # to use the members of A038109 <= N
P:= select(isprime, [$1..floor(sqrt(N))]):
S:= {}:
for p in P do
  Ks:= select(t -> t mod p <> 0, {$1..floor(N/p^2)});
  R:= map(`*`,Ks,p^2) minus S;
  for r in R do B[r]:= p od:
  S:= S union R;
od:
A038109:= sort(convert(S,list)):
seq(B[A038109[i]], i=1..nops(A038109)); # Robert Israel, Mar 28 2017

s[n_] := If[(pos = Position[(f = FactorInteger[n])[[;; , 2]], 2]) == {}, 1, f[[pos[[1, 1]], 1]]]; Select[Array[s, 300], # > 1 &] (* Amiram Eldar, Nov 14 2020 *)

a(n) = sqrt(A284018(n)). - Amiram Eldar, Nov 14 2020

Corrected by Robert Israel, Mar 28 2017

cp's OEIS Frontend

A038109 Divisible exactly by the square of a prime.

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