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

1, 2, 4, 6, 8, 12, 14, 16, 24, 32, 64, 128, 152, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 1

Author

Omar E. Pol, May 16 2016

Keywords

• nonn

Comments

All powers of 2 are in the sequence because the bottom entries of their difference triangles are always 1's.

Besides 6, 12, 14, 24 and 152, are there any other non-powers of 2 in this sequence? - David A. Corneth, May 19 2016

Examples

			For n = 14 the difference triangle of the divisors of 14 is
1 . 2 . 7 . 14
. 1 . 5 . 7
. . 4 . 2
. . .-2
The bottom entry is -2 and -2 divides 14, so 14 is in the sequence.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..40

Crossrefs

Cf. A000079, A027750, A187202, A273102, A273103, A273109.

Programs

Mathematica

Select[Range[10^6], Function[k, If[k == {0}, False, Divisible[#, First@ k]]]@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &] (* Michael De Vlieger, May 17 2016 *)

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

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

Sage

def is_A273131(n):
    D = divisors(n)
    T = matrix(ZZ, len(D))
    for m, d in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return T[len(D)-1, 0].divides(n)
print([n for n in range(1, 6000) if is_A273131(n)])
# Peter Luschny, May 18 2016

Extensions

a(12) = 128 and a(14)-a(25) from Michel Marcus, May 16 2016

a(26)-a(28) from David A. Corneth, May 19 2016

a(29)-a(37) from Lars Blomberg, Oct 18 2016