A002791 a(n) = Sum_{d|n, d <= 4} d^2 + 4*Sum_{d|n, d>4} d.

1, 5, 10, 21, 21, 38, 29, 53, 46, 65, 45, 102, 53, 89, 90, 117, 69, 146, 77, 161, 122, 137, 93, 230, 121, 161, 154, 217, 117, 278, 125, 245, 186, 209, 189, 354, 149, 233, 218, 353, 165, 374, 173, 329, 306, 281, 189, 486, 225, 365, 282, 385, 213, 470, 285, 473, 314, 353, 237, 662, 245, 377, 410, 501, 333, 566, 269, 497

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• 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

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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.

Crossrefs

Cf. A002659, A002660.

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(4);

Mathematica

a[n_] := DivisorSum[n, #^2 &, # < 5 &] + 4 * DivisorSum[n, # &, # > 4 &]; Array[a, 70] (* Amiram Eldar, Aug 17 2019 *)

Formula

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

Conjectured: a(n) = 4 * sigma(n) - f(n mod 6) where f(0) = 10, f(1) = 3, f(2) = 7, f(3) = 6, f(4) = 7, f(5) = 3. - Sean A. Irvine, May 17 2014

Extensions

Edited by N. J. A. Sloane, May 21 2014