

A187202


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


25



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
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OFFSET

1,3


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
. . ba
b . . . . c2b+a
. . cb . . . . . d3c+3ba
c . . . . d2c+b . . . . . . e4d+6c4b+a
. . dc . . . . . e3d+3cb
d . . . . e2d+c
. . ed
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

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


FORMULA

a(n) = Sum_{k=0..d1} (1)^k*binomial(d1,k)*D[dk], 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.


EXAMPLE

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.


MAPLE

f:=proc(n) local k, d, lis; lis:=divisors(n); d:=nops(lis);
add( (1)^k*binomial(d1, k)*lis[dk], k=0..d1); 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 *)


PROG

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


CROSSREFS

Cf. A000005, A007318, A027750, A187203, A273102.
Sequence in context: A249151 A046791 A187203 * A125131 A003958 A326140
Adjacent sequences: A187199 A187200 A187201 * A187203 A187204 A187205


KEYWORD

easy,sign


AUTHOR

Omar E. Pol, Aug 01 2011


EXTENSIONS

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


STATUS

approved



