A347173 - OEIS

%I #19 Feb 24 2024 10:16:07

%S 1,1,1,1,1,1,1,1,10,1,1,10,1,1,10,1,1,10,1,1,10,1,1,10,26,1,10,1,1,35,

%T 1,1,10,1,26,10,1,1,10,26,1,10,1,1,35,1,1,10,50,26,10,1,1,10,26,50,10,

%U 1,1,35,1,1,59,1,26,10,1,1,10,75,1,10,1,1,35,1,50,10,1,26

%N Sum of squares of odd divisors of n that are <= sqrt(n).

%H David A. Corneth, <a href="/A347173/b347173.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} (2*k - 1)^2 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).

%e a(18) = 10 as the odd divisors of 18 are the divisors of 9 which are 1, 3 and 9. Of those, 1 and 3 are <= sqrt(18) so we find the squares of 1 and 3 then add them i.e., a(18) = 1^2 + 3^2 = 10. - _David A. Corneth_, Feb 24 2024

%t Table[DivisorSum[n, #^2 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 80}]

%t nmax = 80; CoefficientList[Series[Sum[(2 k - 1)^2 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n) = sum(k=0, sqrtint(n), if ((k%2) && !(n%k), k^2)); \\ _Michel Marcus_, Aug 22 2021

%o (PARI)

%o a(n) = {

%o 	my(s = sqrtint(n), res);

%o 	n>>=valuation(n, 2);

%o 	d = divisors(n);

%o 	for(i = 1, #d,

%o 		if(d[i] <= s,

%o 			res += d[i]^2

%o 		,

%o 			return(res)

%o 		)

%o 	); res

%o } \\ _David A. Corneth_, Feb 24 2024

%Y Cf. A001157, A050999, A069288, A069289, A095118, A347161, A347174, A347175.

%K nonn,easy

%O 1,9

%A _Ilya Gutkovskiy_, Aug 21 2021