A072502 Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways.

9, 18, 25, 36, 49, 50, 72, 98, 100, 121, 144, 169, 196, 200, 242, 288, 289, 338, 361, 392, 400, 484, 529, 576, 578, 676, 722, 784, 800, 841, 961, 968, 1058, 1152, 1156, 1352, 1369, 1444, 1568, 1600, 1681, 1682, 1849, 1922, 1936, 2116, 2209, 2304, 2312, 2704

Offset: 1

Author

Ron Knott, Jan 27 2003

Keywords

• easy
• nonn

Comments

Also numbers that are the product of a power of 2 (A000079) and the square of an odd prime, or numbers having exactly 3 odd divisors: A001227(a(n)) = 3. - Reinhard Zumkeller, May 01 2012

Numbers n such that the symmetric representation of sigma(n) has 3 subparts. - Omar E. Pol, Dec 28 2016

Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 2 ways. E.g., 2+3+4 = 9 and 4+5 = 9, 3+4+5+6 = 18 and 5+6+7 = 18. - Julie Jones, Aug 13 2018

Appears to be numbers n such that tau(2*n) = tau(n) + 3. - Gary Detlefs, Jan 22 2020

Column 3 of A266531. - Omar E. Pol, Dec 01 2020

Examples

			a(1)=9 is the smallest number with 3 run sums: 2+3+4 = 4+5 = 9.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Ron Knott, Introducing Runsums - a sum of consecutive integers.
• T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article #99.1.6.

Crossrefs

Not to be confused with A069562.

Cf. A001227, A001248, A038547, A038550, A085548, A237593, A266531, A279387.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a072502 n = a072502_list !! (n-1)
a072502_list = f (singleton 9) $ drop 2 a001248_list where
   f s (x:xs) = m : f (insert (2 * m) $ insert x s') xs where
                (m,s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012

Formula

Sum_{n>=1} 1/a(n) = 2 * Sum_{p odd prime} 1/p^2 = 2 * A085548 - 1/2 = 0.404494... - Amiram Eldar, Feb 18 2021

Extensions

Extended by Ray Chandler, Dec 30 2011