A002660 a(n) = Sum_{d|n, d <= 3} d^2 + 3*Sum_{d|n, d>3} d. (Formerly M3797 N1549)

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

1,2

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), 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).

Links

Table of n, a(n) for n=1..69.

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

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 *)

Prog

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

Crossrefs

Cf. A000203, A002659, A002791.

A row of the array in A242639.

Sequence in context: A054514 A372624 A200940 * A367021 A313941 A313942

Adjacent sequences: A002657 A002658 A002659 * A002661 A002662 A002663

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved