cp's OEIS Frontend

A382767 Smallest number k that is powerful but not a prime power that is also coprime to n.

%I A382767 #8 Apr 05 2025 10:58:23
%S A382767 36,225,100,225,36,1225,36,225,100,441,36,1225,36,225,196,225,36,1225,
%T A382767 36,441,100,225,36,1225,36,225,100,225,36,5929,36,225,100,225,36,1225,
%U A382767 36,225,100,441,36,3025,36,225,196,225,36,1225,36,441,100,225,36,1225
%N A382767 Smallest number k that is powerful but not a prime power that is also coprime to n.
%C A382767 Let p be the smallest prime that is coprime to n and let q be the second smallest prime that is coprime to n. Then a(n) = p^2 * q^2.
%C A382767 Records in this sequence are set by n in A002110.
%H A382767 Michael De Vlieger, <a href="/A382767/b382767.txt">Table of n, a(n) for n = 1..10000</a>
%F A382767 a(n) = A053669(n)^2 * A380539(n)^2.
%F A382767 a(n) = A381805(n)^2.
%F A382767 a(n) = (A382248(n)/A020639(n))^2.
%F A382767 For k and m such that rad(k) = rad(m), a(k) = a(m), where rad = A007947.
%e A382767 a(1) = 36 = (2*3)^2, since p = 2, q = 3.
%e A382767 a(2) = 225 = (3*5)^2, since p = 3, q = 5.
%e A382767 a(3) = 100 = (2*5)^2, since p = 2, q = 5.
%e A382767 a(4) = 225 = (3*5)^2, since p = 3, q = 5, a(2^i) = 225 for i > 0.
%e A382767 a(6) = 1225 = (5*7)^2, since p = 5, q = 7.
%e A382767 a(9) = 400 = (2*5)^2, since p = 2, q = 5, a(3^i) = 100 for i > 0.
%e A382767 a(10) = 441 = (3*7)^2, since p = 3, q = 7.
%e A382767 a(12) = 1225 = (5*7)^2, since p = 5, q = 7, a(k) = 1225 for n in A033845 (i.e., n such that rad(n) = 6), where rad = A007947.
%e A382767 a(20) = 441 = (3*7)^2, since p = 3, q = 7, a(k) = 441 for n in A033846 (i.e., n such that rad(n) = 10).
%e A382767 a(30) = 5929 = (7*11)^2, since p = 7, q = 11, etc.
%t A382767 Table[c = 0; q = 2; Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]
%Y A382767 Cf. A002110, A007947, A053669, A286708, A380539, A381805, A382248.
%K A382767 nonn,easy
%O A382767 1,1
%A A382767 _Michael De Vlieger_, Apr 04 2025