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

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

1,4

Comments

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)

Links

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

Formula

a(n) = n - A187202(n).

a(n) = 1, if n is prime.

a(2^k) = 2^k - 1 = A000225(k), k >= 0.

Example

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.

Mathematica

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

Prog

(Sage)
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
(PARI) 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

Crossrefs

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

Sequence in context: A326065 A078896 A322582 * A318841 A300238 A180062

Adjacent sequences: A273130 A273131 A273132 * A273134 A273135 A273136

Keyword

sign

Author

Omar E. Pol, May 17 2016

Status

approved