This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A359390 #54 Jan 28 2023 12:17:00
%S A359390 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,36,37,41,43,47,49,
%T A359390 53,59,61,64,67,71,73,79,81,83,89,97,100,101,103,107,109,113,121,125,
%U A359390 127,128,131,137,139,144,149,151,157,163,167,169,173,179,181,191,193
%N A359390 Sequence lists the numbers k such that bottom entry is an integer in the ratio d(i+1)/d(i) triangle of the elements in the divisors of n, where d(1) < d(2) < ... < d(q) denote the divisors of k.
%C A359390 The corresponding integer bottom entry is 1 if k is nonprime or k if k is prime. [It is very likely that this is true, but no proof has yet been given. - _Jianing Song_, Jan 22 2023]
%C A359390 We observe that a(n) = A323306(n) for n = 1..50. But a(51) = 144 does not belong to that sequence.
%C A359390 Note that the bottom rational is Product_{i=1..q} d(i) ^ (binomial(q-1,i-1) * (-1)^(q-i)). - _Kevin Ryde_, Jan 03 2023
%C A359390 Given n, let 1 = M(1,1) < M(1,2) < ... < M(1,d) = n be the divisors of n, and M(i,j) = M(i-1,j+1)/M(i-1,j) for 2 <= i <= d, 1 <= j <= d+1-i. Since M(1,d+1-j) = n/M(1,j) for 1 <= j <= d, we have M(i,d+2-i-j) = M(i,j) for even i, 1 <= j <= d+1-i, and M(i,d+2-i-j) = 1/M(i,j) for odd i > 1, 1 <= j <= d+1-i. If n is a square, then d is odd, so M(d,1) = 1/M(d,1) => M(d,1) = 1. This shows that all square numbers are terms. Note that all powers of primes (A000961) are trivially terms. It seems that the squares and the powers of primes are the only terms. - _Jianing Song_, Jan 03 2023
%e A359390 36 is a term because the triangle of the elements d(i+1)/d(i) has bottom entry 1:
%e A359390 [1, 2, 3, 4, 6, 9, 12, 18, 36]
%e A359390 [2, 3/2, 4/3, 3/2, 3/2, 4/3, 3/2, 2]
%e A359390 [3/4, 8/9, 9/8, 1, 8/9, 9/8, 4/3]
%e A359390 [32/27, 81/64, 8/9, 8/9, 81/64, 32/27]
%e A359390 [2187/2048, 512/729, 1, 729/512, 2048/2187]
%e A359390 [1048576/1594323, 729/512, 729/512, 1048576/1594323]
%e A359390 [1162261467/536870912, 1, 536870912/1162261467]
%e A359390 [536870912/1162261467, 536870912/1162261467]
%e A359390 [1].
%e A359390 6 is not a term because the triangle of the elements d(i+1)/d(i) has bottom entry 16/9.
%e A359390 [1, 2, 3, 6]
%e A359390 [2, 3/2, 2]
%e A359390 [3/4, 4/3]
%e A359390 [16/9]
%t A359390 Lst={}; Table[d=Divisors[n]; While[Length[d]>1,d=Ratios[d]]; If[d[[1]]==Floor[d[[1]]],AppendTo[Lst,n]],{n,300}]; Lst
%o A359390 (PARI) isA359390(n) = my(L = factor(n), w = #L~, v=divisors(n), q=#v); for(i_d=1, q-1, for(i_p=1, w, L[i_p,2] += binomial(q-1,i_d-1) * (-1)^(q-i_d) * valuation(v[i_d], L[i_p,1]))); for(i_p=1, w, if(L[i_p,2]<0, return(0))); return(1) \\ _Jianing Song_, Jan 22 2023, based on the formula provided by _Kevin Ryde_
%Y A359390 Cf. A323306. Contains A000290 and A000961 as subsequences (and conjectured to be the union of these two sequences).
%K A359390 nonn
%O A359390 1,2
%A A359390 _Michel Lagneau_, Jan 03 2023