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%I #9 May 21 2014 18:04:54
%S 1,3,5,4,7,10,7,13,17,21,6,11,16,21,26,12,23,32,38,44,50,8,15,22,29,
%T 36,43,50,15,29,41,53,61,69,77,85,13,25,37,46,55,64,73,82,91,18,35,50,
%U 65,80,90,100,110,120,130,12,23,34,45,56,67,78,89,100,111,122,28,55,80,102,120,138,150,162,174,186,198,210
%N Triangle read by rows: T(s,n) (1 <= s <= n) = Sum_{d|n, d <= s} d^2 + s*Sum_{d|n, d>s} d.
%D P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. Note that the entry 53 should be 50.
%e Triangle begins:
%e [1]
%e [3, 5]
%e [4, 7, 10]
%e [7, 13, 17, 21]
%e [6, 11, 16, 21, 26]
%e [12, 23, 32, 38, 44, 50]
%e [8, 15, 22, 29, 36, 43, 50]
%e [15, 29, 41, 53, 61, 69, 77, 85]
%e ...
%e The full array (see A242639) begins:
%e 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...
%e 1, 5, 7, 13, 11, 23, 15, 29, 25, 35, 23, 55, ...
%e 1, 5, 10, 17, 16, 32, 22, 41, 37, 50, 34, 80, ...
%e 1, 5, 10, 21, 21, 38, 29, 53, 46, 65, 45, 102, ...
%e 1, 5, 10, 21, 26, 44, 36, 61, 55, 80, 56, 120, ...
%e 1, 5, 10, 21, 26, 50, 43, 69, 64, 90, 67, 138, ...
%e 1, 5, 10, 21, 26, 50, 50, 77, 73, 100, 78, 150, ...
%e 1, 5, 10, 21, 26, 50, 50, 85, 82, 110, 89, 162, ...
%e ...
%p with(numtheory):
%p A:=proc(s,n) local d,s1,s2;
%p s1:=0; s2:=0;
%p for d in divisors(n) do
%p if d <= s then s1:=s1+d^2 else s2:=s2+d; fi; od:
%p s1+s*s2; end;
%p for n from 1 to 15 do lprint([seq(A(s,n),s=1..n)]); od:
%Y Upper triangle of array in A242639.
%K nonn,tabl
%O 1,2
%A _N. J. A. Sloane_, May 21 2014
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