cp's OEIS Frontend

A381805 Smallest composite squarefree number that is coprime to n.

%I A381805 #7 Apr 05 2025 10:58:12
%S A381805 6,15,10,15,6,35,6,15,10,21,6,35,6,15,14,15,6,35,6,21,10,15,6,35,6,15,
%T A381805 10,15,6,77,6,15,10,15,6,35,6,15,10,21,6,55,6,15,14,15,6,35,6,21,10,
%U A381805 15,6,35,6,15,10,15,6,77,6,15,10,15,6,35,6,15,10,33,6,35
%N A381805 Smallest composite squarefree number that is coprime to n.
%C A381805 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*q.
%C A381805 Records in this sequence are set by n in A002110.
%H A381805 Michael De Vlieger, <a href="/A381805/b381805.txt">Table of n, a(n) for n = 1..10000</a>
%F A381805 a(n) = A053669(n) * A380539(n) = A382248(n)/A020639(n).
%F A381805 For k and m such that rad(k) = rad(m), a(k) = a(m), where rad = A007947.
%F A381805 n < a(n) for n in A051250, a finite sequence whose largest term is 60.
%e A381805 a(1) = 6 = 2*3, since p = 2, q = 3.
%e A381805 a(2) = 15 = 3*5, since p = 3, q = 5.
%e A381805 a(3) = 10 = 2*5, since p = 2, q = 5.
%e A381805 a(4) = 15 = 3*5, since p = 3, q = 5, a(2^i) = 15 for i > 0.
%e A381805 a(6) = 35 = 5*7, since p = 5, q = 7.
%e A381805 a(9) = 20 = 2*5, since p = 2, q = 5, a(3^i) = 10 for i > 0.
%e A381805 a(10) = 21 = 3*7, since p = 3, q = 7.
%e A381805 a(12) = 35 = 5*7, since p = 5, q = 7, a(k) = 35 for n in A033845 (i.e., n such that rad(n) = 6).
%e A381805 a(20) = 21 = 3*7, since p = 3, q = 7, a(k) = 21 for n in A033846 (i.e., n such that rad(n) = 10).
%e A381805 a(30) = 77 = 7*11, since p = 7, q = 11, etc.
%t A381805 Table[c = 0; q = 2; Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]
%o A381805 (PARI) a(n) = my(k=2); while (isprime(k) || !issquarefree(k) || (gcd(k, n) != 1) , k++); k; \\ _Michel Marcus_, Apr 01 2025
%Y A381805 Cf. A002110, A007947, A051250, A053669, A120944, A380539, A382248.
%K A381805 nonn,easy
%O A381805 1,1
%A A381805 _Michael De Vlieger_, Mar 31 2025