%I #41 Feb 23 2019 08:00:34

%S 1,2,4,8,12,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,

%T 65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,

%U 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592

%N Numbers n such that in the difference triangle of the divisors of n (including the divisors of n) the diagonal from the bottom entry to n gives the divisors of n.

%C Is this also the union of 12 and the powers of 2?

%C All powers of 2 are in the sequence.

%e For n = 12 the difference triangle of the divisors of 12 is

%e 1 . 2 . 3 . 4 . 6 . 12

%e . 1 . 1 . 1 . 2 . 6

%e . . 0 . 0 . 1 . 4

%e . . . 0 . 1 . 3

%e . . . . 1 . 2

%e . . . . . 1

%e The bottom entry is 1, and the diagonal from the bottom entry to 12 is [1, 2, 3, 4, 6, 12] hence the diagonal gives the divisors of 12, so 12 is in the sequence.

%e Note that for n = 12 and the powers of 2 the descending diagonals, from left to right, are symmetrics, for example: the first diagonal is 1, 1, 0, 0, 1, 1.

%t aQ[n_] := Module[{d=Divisors[n]}, nd = Length[d]; vd = d; ans = True; Do[ vd = Differences[vd]; If[Max[vd] != d[[nd-k]], ans=False; Break[]], {k,1,nd-1}]; ans]; Select[Range[100000], aQ] (* _Amiram Eldar_, Feb 23 2019 *)

%o (PARI) isok(n) = {my(d = divisors(n)); my(nd = #d); my(vd = d); for (k=1, nd-1, vd = vector(#vd-1, j, vd[j+1] - vd[j]); if (vecmax(vd) != d[nd-k], return (0));); return (1);} \\ _Michel Marcus_, May 16 2016

%Y Cf. A000079, A027750, A187202, A273102, A273103.

%K nonn

%O 1,2

%A _Omar E. Pol_, May 15 2016

%E a(12)-a(21) from _Michel Marcus_, May 16 2016

%E a(22)-a(35) from _Amiram Eldar_, Feb 23 2019