Average Of Changes Between Number Of Divisors

[Maximilian Hasler]

This could also be described as:
 partial sums of | A051950 |
("starting with n=1", i.e., more precisely,
a(n) = sum(i=2..n, | d(i)-d(i-1) | ) = sum(i=1..n-1, | d(i+1)-d(i) | )
= sum(i=1..n-1, | A051950(i) | )

It might have been more natural to attribute offset 0 to this sequence
(then it would be a(n) = sum( i=1..n, | A051950(i) | ),
while A051950(0) = 1 seems (at least) a bit artificial to me...

More terms and
(PARI)
print1(s=0);for(i=2,99,print1(", ",s+=abs(numdiv(i)-numdiv(i-1))))

0, 1, 1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 21, 23, 23, 24, 27, 31, 35,
39, 41, 41, 43, 49, 54, 55, 55, 57, 61, 67, 73, 77, 79, 79, 79, 84,
91, 93, 93, 97, 103, 109, 115, 119, 119, 121, 123, 131, 138, 141, 143,
145, 149, 155, 159, 163, 167, 167, 169, 179, 189, 191, 193, 194, 197,
201, 207, 211, 213, 217, 223, 233, 243, 245, 247, 247, 249, 253, 259

not very intelligent "direct" PARI function:
a(n)=sum(i=2,n,abs(numdiv(i)-numdiv(i-1)))

check:
gp > vector(10,i,a(i))
% = [0, 1, 1, 2, 3, 5, 7, 9, 10, 11]

On Mon, Apr 21, 2008 at 11:34 AM, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
> I just posted this: (I fix a typo here.)
>
>  %I A000001
>  %S A000001
>  0,1,1,2,3,5,7,9,10,11,13,17,21,23,23,24,27,31,35,39,41,41,43,49,54,55,55,57,61,67
>  %N A000001 a(1)=0. For n >= 2, a(n) = a(n-1) +
>  |d(n)-d(n-1)|, where d(n) is the number of
>  positive divisors of n.
>  %C A000001 For any given n >= 2, a(n)/(n-1) is
>  the average of the |d(k)-d(k-1)|'s among all k
>  where 2<=k<=n.
>  %Y A000001 A000005,A051950
>  %O A000001 1
>  %K A000001 ,more,nonn,