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A273131 Numbers n such that the bottom entry of the difference table of the divisors of n divides n. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A249721 A010063 A260652

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

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Last modified August 8 08:27 EDT 2020. Contains 336293 sequences. (Running on oeis4.)