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

%I A224834 #23 Aug 29 2023 04:19:28
%S A224834 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,
%T A224834 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,
%U A224834 1,5,9,30,5,25,1,14,5,13,1,50,1,5,9
%N A224834 a(n) = Sum {d|n, d <= n^(1/2)} tau(d)^2.
%H A224834 Robert Israel, <a href="/A224834/b224834.txt">Table of n, a(n) for n = 1..10000</a>
%F A224834 If p is prime, a(p^k) = A000330(1+floor(k/2)). - _Robert Israel_, Nov 30 2016
%p A224834 f:= proc(n) add(numtheory:-tau(d)^2, d = select(t -> (t^2<=n), numtheory:-divisors(n))) end proc:
%p A224834 map(f, [$1..100]); # _Robert Israel_, Nov 30 2016
%t A224834 a[n_] := DivisorSum[n, DivisorSigma[0, #]^2 &, #^2 <= n &]; Array[a, 100] (* _Amiram Eldar_, Aug 29 2023 *)
%o A224834 (PARI) a(n) = sumdiv(n, d, (d<=sqrtn(n, 2))*numdiv(d)^2) \\ _Michel Marcus_, Jul 21 2013
%o A224834 (PARI) a(n)=my(s=sqrtint(n)); sumdiv(n,d,if(d<=s,numdiv(d)^2)) \\ _Charles R Greathouse IV_, Jul 22 2013
%Y A224834 Cf. A000330, A007425, A062367, A224835.
%K A224834 nonn
%O A224834 1,4
%A A224834 _Michel Marcus_, Jul 21 2013