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
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
KEYWORD
sign
AUTHOR
Omar E. Pol, May 17 2016
STATUS
approved