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A307252 Records in A319100.

%I A307252 #18 Apr 19 2019 11:29:20
%S A307252 1,2,6,12,24,36,48,72,144,216,288,432,864,1296,1728,2592,5184,7776,
%T A307252 10368,15552,31104,46656,62208,93312,186624,373248,559872,1119744,
%U A307252 2239488,3359232,4478976,6718464,13436928,20155392,26873856,40310784,80621568,120932352
%N A307252 Records in A319100.
%C A307252 All terms are of the form 6^u*2^j. Other than the term 48, k = 6^i*2^j is a term if and only if for all i', j' such that F(i',j') < F(i,j) we have 6^i'*2^j' < 6^i*2^j, where F(i,j) = Product_{s=1..i} (p_s)*Product_{t=1..j} (q_t), where p_1 = 7, p_2 = 9, p_s = A002476(s-1) for s >= 3; q_1 = 4, q_2 = 2, q_t = A007528(t-2) for t >= 3. Or equivalently: (a) for any u, v such that u <= i and 6^u < 2^v, Product_{s=i-u+1..i} (p_s) < Product_{t=j+1..j+v} (q_t); (b) for any u, v such that v <= j and 6^u > 2^v, Product_{s=i+1..i+u} (p_s) > Product_{t=j-v+1..j} (q_t). For example, 746496 = 6^6*2^4 is not a term because (q_3)*(q_4) = 5*11 > p_7 = 43.
%e A307252 A319100(168) = 48 which is larger than A319100(i) for i < 168, so 48 is a term.
%o A307252 (PARI) P(n) = if(!n, 1, if(n==1, 7, my(i=0,N=9); forprime(p=7, oo, if(p%3==1, i++; N*=p); if(i==n-1, return(N)))))
%o A307252 Q(n) = if(!n, 1, if(n==1, 4, my(i=0,N=4); forprime(p=2, oo, if(p%3==2, i++; N*=p); if(i==n-1, return(N)))))
%o A307252 v = []; for(i=0, 15, for(j=0, 15, if(P(i)*Q(j) < min(P(16), Q(16)), v=concat(v, [P(i)*Q(j)])))); v=vecsort(v);
%o A307252 u = []; for(i=1, #v, if(sum(j=1, i-1, A319100(v[j]) >= A319100(v[i]))==0, u=concat(u, [A319100(v[i])])));
%o A307252 vecsort(concat(u, [48])) \\ See A319100 for its program
%Y A307252 Cf. A319100, A307251, A002476, A007528, A025610.
%K A307252 nonn
%O A307252 1,2
%A A307252 _Jianing Song_, Mar 31 2019