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

%I A072502 #50 Feb 18 2021 02:36:10
%S A072502 9,18,25,36,49,50,72,98,100,121,144,169,196,200,242,288,289,338,361,
%T A072502 392,400,484,529,576,578,676,722,784,800,841,961,968,1058,1152,1156,
%U A072502 1352,1369,1444,1568,1600,1681,1682,1849,1922,1936,2116,2209,2304,2312,2704
%N A072502 Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways.
%C A072502 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
%C A072502 Numbers n such that the symmetric representation of sigma(n) has 3 subparts. - _Omar E. Pol_, Dec 28 2016
%C A072502 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
%C A072502 Appears to be numbers n such that tau(2*n) = tau(n) + 3. - _Gary Detlefs_, Jan 22 2020
%C A072502 Column 3 of A266531. - _Omar E. Pol_, Dec 01 2020
%H A072502 Reinhard Zumkeller, <a href="/A072502/b072502.txt">Table of n, a(n) for n = 1..10000</a>
%H A072502 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/index.html">Introducing Runsums - a sum of consecutive integers</a>.
%H A072502 T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, Vol. 2 (1999), Article #99.1.6.
%F A072502 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
%e A072502 a(1)=9 is the smallest number with 3 run sums: 2+3+4 = 4+5 = 9.
%o A072502 (Haskell)
%o A072502 import Data.Set (singleton, deleteFindMin, insert)
%o A072502 a072502 n = a072502_list !! (n-1)
%o A072502 a072502_list = f (singleton 9) $ drop 2 a001248_list where
%o A072502    f s (x:xs) = m : f (insert (2 * m) $ insert x s') xs where
%o A072502                 (m,s') = deleteFindMin s
%o A072502 -- _Reinhard Zumkeller_, May 01 2012
%Y A072502 Not to be confused with A069562.
%Y A072502 Cf. A001227, A001248, A038547, A038550, A085548, A237593, A266531, A279387.
%K A072502 easy,nonn
%O A072502 1,1
%A A072502 _Ron Knott_, Jan 27 2003
%E A072502 Extended by _Ray Chandler_, Dec 30 2011