A224914 Partial sums of A217854.

-1, 3, 12, -52, -27, 1269, 1318, 5414, 4685, 14685, 14806, 3000790, 3000959, 3039375, 3090000, 2041424, 2041713, 36053937, 36054298, 100054298, 100248779, 100483035, 100483564, 110175797740, 110175782115, 110176239091, 110176770532, 110658660836

Offset: 1

Author

Simon Jensen, Apr 19 2013

Keywords

• sign

Comments

If there is some n > 47 such that a(n) < 0, then there is some k^2 > 47 such that a(k^2) < 0.

If n > 1 is a square number, then a(n) = a(n-1) - n^tau(n).

If n > 1 is a nonsquare number, then a(n) = a(n-1) + n^tau(n).

If n > 1 is a prime, then a(n) = a(n-1) + n^2.

Examples

			a(4) = a(1) + a(2) + a(3) + (-4)^tau(4) = (-1) + 3 + 12 + (-64) = -52.
		

Links

• Simon Jensen, Table of n, a(n) for n = 1..10000
• Simon Jensen, On an extended divisor product summatory function

Programs

Mathematica

Accumulate@ Table[(-n)^DivisorSigma[0, n], {n, 28}] (* Michael De Vlieger, Mar 18 2016 *)

PARI

a(n) = sum(k=1, n, (-k)^numdiv(k)); \\ Michel Marcus, Mar 18 2016

Formula

a(n) = Sum_{i=1..n} (-i)^tau(i) = Sum_{i=1..n} (-i)^A000005(i) = Sum_{i=1..n} A217854(i).