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

1,2

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A043756 A043765 A043569 * A357371 A360971 A249721

Adjacent sequences: A273128 A273129 A273130 * A273132 A273133 A273134

Keyword

nonn

Author

Omar E. Pol, May 16 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

Status

approved