cp's OEIS Frontend

A359493 Numbers k such that the bottom entry in the ratio d(i)/d(i+1) triangle of the elements in the divisors of n, where d(1) < d(2) < ... < d(q) denote the divisors of k, is equal to 1.

%I A359493 #24 Feb 12 2023 10:38:22
%S A359493 1,4,8,9,16,25,27,32,36,49,64,81,100,121,125,128,144,169,196,225,243,
%T A359493 256,289,324,343,361,400,441,484,512,529,576,625,676,729,784,841,900,
%U A359493 961,1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681,1764,1849,1936
%N A359493 Numbers k such that the bottom entry in the ratio d(i)/d(i+1) triangle of the elements in the divisors of n, where d(1) < d(2) < ... < d(q) denote the divisors of k, is equal to 1.
%C A359493 Subsequence of A001597, but A001597(20) = 216, A001597(41) = 1000, A001597(53) = 1728 are not here.
%C A359493 Observation according to a comment by Bernard Schott: 216, 1000, 1728, 2744, 5832, ... are the first terms of A124581 (abundant cubes), hence the following conjecture: terms in A124581 do not belong to this sequence.
%e A359493 100 is a term because the d(i)/d(i+1) triangle has bottom entry 1:
%e A359493   [1, 2, 4, 5, 10, 20, 25, 50, 100]
%e A359493   [1/2, 1/2,  4/5, 1/2, 1/2, 4/5, 1/2, 1/2]
%e A359493   [1, 5/8, 8/5, 1, 5/8, 8/5, 1]
%e A359493   [8/5, 25/64, 8/5, 8/5, 25/64, 8/5]
%e A359493   [512/125, 125/512, 1, 512/125, 125/512]
%e A359493   [262144/15625, 125/512, 125/512, 262144/15625]
%e A359493   [134217728/1953125, 1, 1953125/134217728]
%e A359493   [134217728/1953125, 134217728/1953125]
%e A359493   [1]
%e A359493 6 is not a term because the d(i)/d(i+1) triangle has bottom entry 9/16.
%e A359493   [1, 2, 3, 6]
%e A359493   [1/2, 2/3, 1/2]
%e A359493   [3/4, 4/3]
%e A359493   [9/16]
%t A359493 Lst={}; Table[d=Divisors[n]; While[Length[d]>1,d=Ratios[Reverse[d]]]; If[d[[1]]==Floor[d[[1]]],AppendTo[Lst,n]],{n,2000}]; Lst
%o A359493 (PARI)
%o A359493 ratios(v) = { my(u=vector(#v-1)); for(i=1,#u,u[i] = v[i]/v[1+i]); (u); };
%o A359493 isA359493(n) = { my(ds=divisors(n)); while(#ds>1, ds = ratios(ds)); (1==ds[1]); }; \\ _Antti Karttunen_, Jan 04 2023
%o A359493 (PARI)  is(n) = { if(!(ispower(n) || n==1), return(0)); my(f = factor(n), d = divisors(f), m = Map(), i, j, nv, e, fd); for(i = 1, #d, e = (-1)^i * binomial(#d-1, i-1); fd = factor(d[i]); for(j = 1, #fd~, if(mapisdefined(m, fd[j, 1]), nv = mapget(m, fd[j, 1]); mapput(m, fd[j, 1], nv + e * fd[j, 2]) , mapput(m, fd[j, 1], e * fd[j, 2]) ) ) ); for(i = 1, #f~, if(mapget(m, f[i, 1]) != 0, return(0) ) ); return(1) } \\ _David A. Corneth_, Jan 07 2023
%Y A359493 Cf. A001597, A359247.
%K A359493 nonn
%O A359493 1,2
%A A359493 _Michel Lagneau_, Jan 03 2023