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.
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936
Offset: 1
Keywords
Examples
100 is a term because the d(i)/d(i+1) triangle has bottom entry 1: [1, 2, 4, 5, 10, 20, 25, 50, 100] [1/2, 1/2, 4/5, 1/2, 1/2, 4/5, 1/2, 1/2] [1, 5/8, 8/5, 1, 5/8, 8/5, 1] [8/5, 25/64, 8/5, 8/5, 25/64, 8/5] [512/125, 125/512, 1, 512/125, 125/512] [262144/15625, 125/512, 125/512, 262144/15625] [134217728/1953125, 1, 1953125/134217728] [134217728/1953125, 134217728/1953125] [1] 6 is not a term because the d(i)/d(i+1) triangle has bottom entry 9/16. [1, 2, 3, 6] [1/2, 2/3, 1/2] [3/4, 4/3] [9/16]
Programs
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Mathematica
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 -
PARI
ratios(v) = { my(u=vector(#v-1)); for(i=1,#u,u[i] = v[i]/v[1+i]); (u); }; isA359493(n) = { my(ds=divisors(n)); while(#ds>1, ds = ratios(ds)); (1==ds[1]); }; \\ Antti Karttunen, Jan 04 2023 -
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
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