A002660 a(n) = Sum_{d|n, d <= 3} d^2 + 3*Sum_{d|n, d>3} d.

1, 5, 10, 17, 16, 32, 22, 41, 37, 50, 34, 80, 40, 68, 70, 89, 52, 113, 58, 122, 94, 104, 70, 176, 91, 122, 118, 164, 88, 212, 94, 185, 142, 158, 142, 269, 112, 176, 166, 266, 124, 284, 130, 248, 232, 212, 142, 368, 169, 275, 214, 290, 160, 356, 214, 356, 238, 266, 178, 500, 184, 284, 310, 377, 250, 428, 202, 374, 286

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A000203, A002659, A002791.

A row of the 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;
f:=s->[seq(A(s,n),n=1..80)]; f(3);

Mathematica

a[n_] := DivisorSum[n, If[# <= 3, #^2, 3 #]&];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 09 2023 *)

PARI

a(n) = sumdiv(n, d, if (d<=3, d^2) + 3*if (d>3, d)); \\ Michel Marcus, Mar 09 2023

Formula

Conjectured: Inverse Moebius transform of g.f.: (x + 2x^2 + 2x^3 - 2x^4) / (1 - x)^2. - Sean A. Irvine, May 16 2014

a(n) = 3 * sigma(n) - k, where k = 4 if n is even and k = 2 if n is odd. - Sean A. Irvine, May 16 2014

Extensions

Entry revised by N. J. A. Sloane, May 21 2014