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

1,3

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.

Links

Table of n, a(n) for n=1..78.

Example

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, ...
...

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 *)

Crossrefs

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

Main diagonal is A001157.

See A242640 for the upper triangle of this array.

Sequence in context: A365671 A200920 A290534 * A117853 A104734 A029655

Adjacent sequences: A242636 A242637 A242638 * A242640 A242641 A242642

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 21 2014

Status

approved