A224834 a(n) = Sum {d|n, d <= n^(1/2)} tau(d)^2.

1, 1, 1, 5, 1, 5, 1, 5, 5, 5, 1, 9, 1, 5, 5, 14, 1, 9, 1, 14, 5, 5, 1, 18, 5, 5, 5, 14, 1, 13, 1, 14, 5, 5, 5, 34, 1, 5, 5, 18, 1, 25, 1, 14, 9, 5, 1, 34, 5, 9, 5, 14, 1, 25, 5, 18, 5, 5, 1, 38, 1, 5, 9, 30, 5, 25, 1, 14, 5, 13, 1, 50, 1, 5, 9

Offset: 1

Author

Michel Marcus, Jul 21 2013

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000330, A007425, A062367, A224835.

Programs

Maple

f:= proc(n) add(numtheory:-tau(d)^2, d = select(t -> (t^2<=n), numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Nov 30 2016

Mathematica

a[n_] := DivisorSum[n, DivisorSigma[0, #]^2 &, #^2 <= n &]; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

PARI

a(n) = sumdiv(n, d, (d<=sqrtn(n, 2))*numdiv(d)^2) \\ Michel Marcus, Jul 21 2013

PARI

a(n)=my(s=sqrtint(n)); sumdiv(n,d,if(d<=s,numdiv(d)^2)) \\ Charles R Greathouse IV, Jul 22 2013

Formula

If p is prime, a(p^k) = A000330(1+floor(k/2)). - Robert Israel, Nov 30 2016