A242639 Array read by antidiagonals upwards: A(s,n) (s>=1, n >= 1) = Sum_{d|n, d <= s} d^2 + s*Sum_{d|n, d>s} d.

1, 1, 3, 1, 5, 4, 1, 5, 7, 7, 1, 5, 10, 13, 6, 1, 5, 10, 17, 11, 12, 1, 5, 10, 21, 16, 23, 8, 1, 5, 10, 21, 21, 32, 15, 15, 1, 5, 10, 21, 26, 38, 22, 29, 13, 1, 5, 10, 21, 26, 44, 29, 41, 25, 18, 1, 5, 10, 21, 26, 50, 36, 53, 37, 35, 12, 1, 5, 10, 21, 26, 50, 43, 61, 46, 50, 23, 28

Offset: 1

Author

N. J. A. Sloane, May 21 2014

Keywords

• nonn
• tabl

Examples

			The array 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, ...
1, 5, 10, 21, 26, 50, 50, 85, 91, 120, 100, 174, ...
1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 111, 186, ...
...
		

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

Rows give A000203, A002659, A002660, A002791, A241603, A242643.

Main diagonal is A001157.

See A242640 for the upper triangle of this array.

Programs

Maple

# Produces the square array:
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 s from 1 to 12 do lprint([seq(A(s,n),n=1..12)]); od:

Mathematica

A[s_, n_] := DivisorSum[n, If[#<=s, #^2, 0]+If[#>s, s*#, 0]&];
Table[A[s-n+1, n], {s, 1, 12}, {n, 1, s}] // Flatten (* Jean-François Alcover, Mar 07 2023 *)