cp's OEIS Frontend

A382248 Smallest number k that is neither squarefree nor a prime power such that k is coprime to n.

%I A382248 #39 Apr 05 2025 10:58:44
%S A382248 12,45,20,45,12,175,12,45,20,63,12,175,12,45,28,45,12,175,12,63,20,45,
%T A382248 12,175,12,45,20,45,12,539,12,45,20,45,12,175,12,45,20,63,12,275,12,
%U A382248 45,28,45,12,175,12,63,20,45,12,175,12,45,20,45,12,539,12,45,20
%N A382248 Smallest number k that is neither squarefree nor a prime power such that k is coprime to n.
%C A382248 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.
%C A382248 Records in this sequence are set by n in A002110.
%H A382248 Michael De Vlieger, <a href="/A382248/b382248.txt">Table of n, a(n) for n = 1..10000</a>
%F A382248 a(n) = A053669(n)^2 * A380539(n).
%F A382248 For k and m such that rad(k) = rad(m), a(k) = a(m), where rad = A007947.
%e A382248 a(1) = 12 = 2^2*3, since p = 2, q = 3.
%e A382248 a(2) = 45 = 3^2*5, since p = 3, q = 5.
%e A382248 a(3) = 20 = 2^2*5, since p = 2, q = 5.
%e A382248 a(4) = 45 = 3^2*5, since p = 3, q = 5, a(2^i) = 45 for i > 0.
%e A382248 a(6) = 175 = 5^2*7, since p = 5, q = 7.
%e A382248 a(9) = 20 = 2^2*5, since p = 2, q = 5, a(3^i) = 20 for i > 0.
%e A382248 a(10) = 63 = 3^2*7, since p = 3, q = 7.
%e A382248 a(12) = 175 = 5^2*7, since p = 5, q = 7, a(k) = 175 for n in A033845 (i.e., n such that rad(n) = 6).
%e A382248 a(20) = 63 = 3^2*7, since p = 3, q = 7, a(k) = 63 for n in A033846 (i.e., n such that rad(n) = 10).
%e A382248 a(30) = 539 = 7^2*11, since p = 7, q = 11, etc.
%t A382248 Table[c = 0; q = 2; Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]
%o A382248 (PARI) a(n) = my(k=2); while (isprimepower(k) || issquarefree(k) || (gcd(k, n) != 1) , k++); k; \\ _Michel Marcus_, Apr 01 2025
%Y A382248 Cf. A002110, A007947, A053669, A126706, A380539.
%K A382248 nonn,easy
%O A382248 1,1
%A A382248 _Michael De Vlieger_, Mar 31 2025