A187202 The bottom entry in the difference table of the divisors of n.

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, -2, 8, 1, 16, 12, 18, -11, 8, -6, 22, -12, 16, -8, 8, -3, 28, 50, 30, 1, 8, -12, 28, -11, 36, -14, 8, -66, 40, 104, 42, 13, 24, -18, 46, -103, 36, -16, 8, 21, 52, 88, 36, 48, 8, -24, 58, -667, 60, -26, -8, 1, 40, 72

Offset: 1

Author

Omar E. Pol, Aug 01 2011

Keywords

• easy
• sign

Comments

Note that if n is prime then a(n) = n - 1.

Note that if n is a power of 2 then a(n) = 1.

a(A193671(n)) > 0; a(A187204(n)) = 0; a(A193672(n)) < 0. [Reinhard Zumkeller, Aug 02 2011]

First differs from A187203 at a(14). - Omar E. Pol, May 14 2016

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)

Examples

			a(18) = 12 because 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
with bottom entry a(18) = 12.
Note that A187203(18) = 4.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A007318, A027750, A187203, A273102.

Programs

Haskell

a187202 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

Maple

f:=proc(n) local k,d,lis; lis:=divisors(n); d:=nops(lis);
add( (-1)^k*binomial(d-1,k)*lis[d-k], k=0..d-1); end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, May 01 2016

Mathematica

Table[d = Divisors[n]; Differences[d, Length[d] - 1][[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)

PARI

A187202(n)={ for(i=2,#n=divisors(n), n=vecextract(n,"^1")-vecextract(n,"^-1")); n[1]}  \\ M. F. Hasler, Aug 01 2011

Formula

a(n) = Sum_{k=0..d-1} (-1)^k*binomial(d-1,k)*D[d-k], where D is a sorted list of the d = A000005(n) divisors of n. - N. J. A. Sloane, May 01 2016

a(2^k) = 1.

Extensions

Edited by N. J. A. Sloane, May 01 2016