A347173 Sum of squares of odd divisors of n that are <= sqrt(n).

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

Offset

1,9

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

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

Example

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

Mathematica

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

Prog

(PARI) a(n) = sum(k=0, sqrtint(n), if ((k%2) && !(n%k), k^2)); \\ Michel Marcus, Aug 22 2021
(PARI)
a(n) = {
	my(s = sqrtint(n), res);
	n>>=valuation(n, 2);
	d = divisors(n);
	for(i = 1, #d,
		if(d[i] <= s,
			res += d[i]^2
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

Crossrefs

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

Sequence in context: A096089 A010176 A347161 * A351647 A168644 A378544

Adjacent sequences: A347170 A347171 A347172 * A347174 A347175 A347176

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Aug 21 2021

Status

approved