cp's OEIS Frontend

A273131 Numbers n such that the bottom entry of the difference table of the divisors of n divides n.

%I A273131 #40 Apr 29 2020 04:33:34
%S A273131 1,2,4,6,8,12,14,16,24,32,64,128,152,256,512,1024,2048,4096,8192,
%T A273131 16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,
%U A273131 8388608,16777216,33554432,67108864,134217728,268435456,536870912,1073741824,2147483648
%N A273131 Numbers n such that the bottom entry of the difference table of the divisors of n divides n.
%C A273131 All powers of 2 are in the sequence because the bottom entries of their difference triangles are always 1's.
%C A273131 Besides 6, 12, 14, 24 and 152, are there any other non-powers of 2 in this sequence? - _David A. Corneth_, May 19 2016
%H A273131 Lars Blomberg, <a href="/A273131/b273131.txt">Table of n, a(n) for n = 1..40</a>
%e A273131 For n = 14 the difference triangle of the divisors of 14 is
%e A273131 1 . 2 . 7 . 14
%e A273131 . 1 . 5 . 7
%e A273131 . . 4 . 2
%e A273131 . . .-2
%e A273131 The bottom entry is -2 and -2 divides 14, so 14 is in the sequence.
%t A273131 Select[Range[10^6], Function[k, If[k == {0}, False, Divisible[#, First@ k]]]@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &] (* _Michael De Vlieger_, May 17 2016 *)
%o A273131 (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]);); vd[1] && ((n % vd[1]) == 0);} \\ _Michel Marcus_, May 16 2016
%o A273131 (PARI) is(n) = my(d=divisors(n),s=sum(i=1,#d,binomial(#d-1,i-1)*(-1)^i*d[i]));if(s!=0,n%s==0) \\ _David A. Corneth_, May 19 2016
%o A273131 (Sage)
%o A273131 def is_A273131(n):
%o A273131     D = divisors(n)
%o A273131     T = matrix(ZZ, len(D))
%o A273131     for m, d in enumerate(D):
%o A273131         T[0, m] = d
%o A273131         for k in range(m-1, -1, -1) :
%o A273131             T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
%o A273131     return T[len(D)-1, 0].divides(n)
%o A273131 print([n for n in range(1, 6000) if is_A273131(n)])
%o A273131 # _Peter Luschny_, May 18 2016
%Y A273131 Cf. A000079, A027750, A187202, A273102, A273103, A273109.
%K A273131 nonn
%O A273131 1,2
%A A273131 _Omar E. Pol_, May 16 2016
%E A273131 a(12) = 128 and a(14)-a(25) from _Michel Marcus_, May 16 2016
%E A273131 a(26)-a(28) from _David A. Corneth_, May 19 2016
%E A273131 a(29)-a(37) from _Lars Blomberg_, Oct 18 2016