A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

1, 3, 5, 8, 9, 14, 13, 20, 21, 26, 21, 35, 25, 38, 41, 48, 33, 57, 37, 64, 61, 62, 45, 84, 65, 74, 81, 96, 57, 109, 61, 112, 101, 98, 101, 138, 73, 110, 121, 151, 81, 160, 85, 160, 161, 134, 93, 196, 133, 185, 161, 192, 105, 216, 173, 223, 181, 170, 117, 258

Offset: 1

Author

Jonas Wallgren, May 16 2008

Keywords

• nonn

Comments

Conjecture: the row sums of the plane partitions A010766 are upper bounds. - R. J. Mathar, Aug 06 2008

a(n) is divisible by n iff n=1 or n belongs to A227993. - Rémy Sigrist, Mar 06 2017

a(n) >= 2*n - 1, with equality iff n is not composite. - Rémy Sigrist, Mar 12 2017

Examples

			a(8) = these 20 marked *'s:
-|12345678
-+--------
1|********
2|****
3|**
4|**
5|*
6|*
7|*
8|*
		

Links

• Paul Tek, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Illustration of the first terms

Crossrefs

Cf. A227993.

Programs

PARI

a(n) = my(ar=0, pw=0); fordiv(n, w, ar=ar+(w-pw)*n/w; pw=w); return (ar) \\ Paul Tek, Mar 21 2015

Formula

a(n) = n*(m - Sum_{k=1..m-1} d(k)/d(k+1)), where d(1) < d(2) < ... < d(m) denote the divisors of n. - Rémy Sigrist, Mar 06 2017

Extensions

More terms from Paul Tek, Mar 21 2015

Typo in name corrected by Rémy Sigrist, Mar 05 2017