A242640 Triangle read by rows: T(s,n) (1 <= s <= n) = Sum_{d|n, d <= s} d^2 + s*Sum_{d|n, d>s} d.

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, 36, 43, 50, 15, 29, 41, 53, 61, 69, 77, 85, 13, 25, 37, 46, 55, 64, 73, 82, 91, 18, 35, 50, 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

Offset: 1

Author

N. J. A. Sloane, May 21 2014

Keywords

• nonn
• tabl

Examples

			Triangle begins:
[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, 36, 43, 50]
[15, 29, 41, 53, 61, 69, 77, 85]
...
The full array (see A242639) begins:
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ...
1, 5, 7, 13, 11, 23, 15, 29, 25, 35, 23, 55, ...
1, 5, 10, 17, 16, 32, 22, 41, 37, 50, 34, 80, ...
1, 5, 10, 21, 21, 38, 29, 53, 46, 65, 45, 102, ...
1, 5, 10, 21, 26, 44, 36, 61, 55, 80, 56, 120, ...
1, 5, 10, 21, 26, 50, 43, 69, 64, 90, 67, 138, ...
1, 5, 10, 21, 26, 50, 50, 77, 73, 100, 78, 150, ...
1, 5, 10, 21, 26, 50, 50, 85, 82, 110, 89, 162, ...
...
		

References

• 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.

Crossrefs

Upper triangle of array in A242639.

Programs

Maple

with(numtheory):
A:=proc(s,n) local d,s1,s2;
s1:=0; s2:=0;
for d in divisors(n) do
if d <= s then s1:=s1+d^2 else s2:=s2+d; fi;  od:
s1+s*s2; end;
for n from 1 to 15 do lprint([seq(A(s,n),s=1..n)]); od: