cp's OEIS Frontend

A369690 a(n) = max(A119288(n), A053669(n)).

%I A369690 #26 Jul 09 2024 08:42:25
%S A369690 2,3,2,3,2,5,2,3,2,5,2,5,2,7,5,3,2,5,2,5,7,11,2,5,2,13,2,7,2,7,2,3,11,
%T A369690 17,7,5,2,19,13,5,2,5,2,11,5,23,2,5,2,5,17,13,2,5,11,7,19,29,2,7,2,31,
%U A369690 7,3,13,5,2,17,23,5,2,5,2,37,5,19,11,5,2,5,2
%N A369690 a(n) = max(A119288(n), A053669(n)).
%C A369690 Equivalently, a(n) is the largest p such that p is the 2nd smallest prime dividing n or the smallest prime not dividing n.
%C A369690 If squarefree n is such that a(n) = p, then a(k) = p for k in the infinite sequence { k = m*n : rad(m) | n }. Consequence of the fact that both A119288(n) and A053669(n) do not depend on multiplicity of prime divisors p | n.
%H A369690 Michael De Vlieger, <a href="/A369690/b369690.txt">Table of n, a(n) for n = 1..10000</a>
%F A369690 a(n) <= A003557(n) for n > 4 in A246547 and for n in A361098.
%F A369690 Numbers n that set records include 1, 2, and squarefree semiprimes, i.e., (A100484 \ {4}) U {1, 2}.
%e A369690 Let p be the second least prime factor of n or 1 if n is a prime power, and let q be the smallest prime that does not divide n.
%e A369690 a(1) = 2 since max(p, q) = max(1, 2) = 2.
%e A369690 a(2) = 3 since max(p, q) = max(1, 3) = 3.
%e A369690 a(4) = 3 since max(p, q) = max(1, 3) = 3.
%e A369690 a(6) = 5 since max(p, q) = max(3, 5) = 5.
%e A369690 a(9) = 2 since max(p, q) = max(1, 2) = 2.
%e A369690 a(15) = 5 since max(p, q) = max(5, 2) = 5.
%e A369690 a(36) = 5 since max(p, q) = max(3, 5) = 5.
%e A369690 Generally,
%e A369690 a(n) = 2 for n in A061345 = union of {1} and sequences { m*p : prime p > 2, rad(m) | p }.
%e A369690 a(n) = 3 for n in A000079 = { 2*m : rad(m) | 2 }.
%e A369690 a(n) = 5 for k in { k = m*d : rad(m) | d, d in {6, 10, 15} }.
%e A369690 a(n) = 7 for k in { k = m*d : rad(m) | d, d in {14, 21, 30, 35} }.
%e A369690 a(n) = 11 for k in { k = m*d : rad(m) | d, d in {22, 33, 55, 77, 210} }, etc.
%t A369690 {2}~Join~Array[If[PrimePowerQ[#],
%t A369690   q = 2; While[Divisible[#, q], q = NextPrime[q]]; q,
%t A369690   q = 2; While[Divisible[#, q], q = NextPrime[q]];
%t A369690     Max[FactorInteger[#][[2, 1]], q]] &, 120, 2]
%Y A369690 Cf. A000079, A002110, A003557, A007947, A024619, A053669, A061345, A096015 (smallest instead of 2nd smallest), A100484, A119288, A246547, A361098.
%K A369690 nonn,easy
%O A369690 1,1
%A A369690 _Peter Munn_ and _Michael De Vlieger_, Feb 18 2024