A273103 -id:A273103 - cp's OEIS frontend

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

Omar E. Pol, May 16 2016

nonn

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

			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.

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

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

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

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

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

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

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

A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 5, 10, 1, 11, 1, 16, 7, 15, 1, 6, 1, 31, 13, 28, 1, 36, 9, 34, 19, 31, 1, -20, 1, 31, 25, 46, 7, 47, 1, 52, 31, 106, 1, -62, 1, 31, 21, 64, 1, 151, 13, 66, 43, 31, 1, -34, 19, 8, 49, 82, 1, 727, 1, 88, 71, 63, 25, -6, 1, 31, 61, 148, 1, 12, 1, 106, 11, 31, 13, 22, 1, 439, 65, 118, 1, 1541

Offset: 1

Omar E. Pol, May 17 2016

sign

From David A. Corneth, May 20 2016: (Start)
The bottom of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is:
1 . 2 . 3 . 6 . 9 . 18
. 1 . 1 . 3 . 3 . 9
. . 0 . 2 . 0 . 6
. . . 2 .-2 . 6
. . . .-4 . 8
. . . . . 12
The bottom entry is 12, so a(18) = 18 - 12 = 6.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A000225, A007318, A187202, A273102, A273103, A273109.

Array[# - First@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &, 84] (* Michael De Vlieger, May 20 2016 *)

a(n) = my(d=divisors(n));n-sum(i=1,#d,binomial(#d-1,i-1)*(-1)^(#d-i)*d[i]) \\ David A. Corneth, May 20 2016

def A273133(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 n - T[len(D)-1, 0]
print([A273133(n) for n in range(1, 85)]) # Peter Luschny, May 18 2016

a(n) = n - A187202(n).
a(n) = 1, if n is prime.
a(2^k) = 2^k - 1 = A000225(k), k >= 0.

A273157 Numbers which have nonpositive entries in the difference table of their divisors (complement of A273130).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 30, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 117

Offset: 1

Peter Luschny, May 16 2016

nonn

Primorial numbers (A002110) greater than 2 are in this sequence.

			30 is in this sequence because the difference table of the divisors of 30 is:
[1, 2, 3, 5, 6, 10, 15, 30]
[1, 1, 2, 1, 4, 5, 15]
[0, 1, -1, 3, 1, 10]
[1, -2, 4, -2, 9]
[-3, 6, -6, 11]
[9, -12, 17]
[-21, 29]
[50]

Cf. A069059, A187202, A273102, A273103, A273109, A273130 (complement).

def nsf(z):
    D = divisors(z)
    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]
            if T[m-k, k] <= 0: return True
    return False
print([n for n in range(1, 100) if nsf(n)])

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

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A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n.

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A273157 Numbers which have nonpositive entries in the difference table of their divisors (complement of A273130).

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Sage