

A242640


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


1



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


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), 113116. Collected Papers, MIT Press, 1978, Vol. I, pp. 13641367. See Table I. Note that the entry 53 should be 50.


LINKS

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


EXAMPLE

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, ...
...


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:


CROSSREFS

Upper triangle of array in A242639.
Sequence in context: A096457 A277897 A082568 * A210195 A069918 A200700
Adjacent sequences: A242637 A242638 A242639 * A242641 A242642 A242643


KEYWORD

nonn,tabl


AUTHOR

N. J. A. Sloane, May 21 2014


STATUS

approved



