A359657
Least k such that A359247(k) = n, or 0 if no such k exists.
Original entry on oeis.org
5, 1, 136, 168, 141, 424, 1867, 680, 3981, 5800, 2216, 13648, 5763, 2728, 8872, 11944, 15752, 6824, 15219, 8352, 17064, 10920, 10400, 38407, 25105, 27304, 36879, 40501, 37077, 20323, 25635, 29073, 57611, 45795, 90197, 61741, 68735, 55319, 46645, 42549, 95412
Offset: 0
a(3) = 168 because the Collatz trajectory of 168 is T = 168 -> 84 -> 42 -> 21 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1 and the absolute difference triangle of the elements of T is:
168 84 42 21 64 32 16 8 4 2 1
84, 42, 21, 43, 32, 16, 8, 4, 2, 1
42, 21, 22, 11, 16, 8, 4, 2, 1
21, 1, 11, 5, 8, 4, 2, 1
20, 10, 6, 3, 4, 2, 1
10, 4, 3, 1, 2, 1
6, 1, 2, 1, 1
5, 1, 1, 0
4, 0, 1
4, 1
3
with bottom entry = A359247(168) = 3.
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nn=20000; Collatz[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Flatten[Table[Collatz[n],{n,nn}]]; Do[k=1; Table[d=Collatz[m]; While[Length[d]>1,d=Abs[Differences[d]]]; If[d[[1]]==u&&k==1,Print[u," ",m];k=0],{m,nn}],{u,0,22}]
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.
Original entry on oeis.org
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
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]
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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
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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
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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
Showing 1-2 of 2 results.
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