A237349 a(n) = Sum_{i=1..n} ( Product_{k|i} d(k) ), where d(n) = A000005(n).

1, 3, 5, 11, 13, 29, 31, 55, 61, 77, 79, 367, 369, 385, 401, 521, 523, 811, 813, 1101, 1117, 1133, 1135, 10351, 10357, 10373, 10397, 10685, 10687, 14783, 14785, 15505, 15521, 15537, 15553, 62209, 62211, 62227, 62243, 71459, 71461, 75557, 75559, 75847, 76135

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2014

Keywords

• nonn
• easy

Comments

Sum of all the products formed by multiplying together the number of divisors of each divisor of the numbers from 1 to n.

Partial sums of A211776. [Joerg Arndt, Feb 11 2014]

Examples

			a(3) = 5. Sum_{i=1..3} ( Product_{k|i} d(k) ) =
( Product_{k|1} d(k) ) + ( Product_{k|2} d(k) ) + ( Product_{k|3} d(k) ) = ( d(1) ) + ( d(1) * d(2) ) + ( d(1) * d(3) ) = 1 + (1)(2) + (1)(2) = 5.
		

Crossrefs

Cf. A000005, A211776.

Programs

Maple

with(numtheory); A237349:=n->add(mul(tau(k)^(1-ceil(i/k)+floor(i/k)), k=1..i), i=1..n); seq(A237349(n), n=1..50);

Mathematica

Table[Sum[Product[DivisorSigma[0, k]^(1-Ceiling[i/k]+Floor[i/k]), {k, i}], {i, n}], {n, 50}]

Formula

a(n) = Sum_{i=1..n} ( Product_{k|i} A000005(k) ).