A347174 Sum of cubes of odd divisors of n that are <= sqrt(n).

1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 28, 1, 1, 28, 126, 1, 28, 1, 1, 153, 1, 1, 28, 1, 126, 28, 1, 1, 28, 126, 1, 28, 1, 1, 153, 1, 1, 28, 344, 126, 28, 1, 1, 28, 126, 344, 28, 1, 1, 153, 1, 1, 371, 1, 126, 28, 1, 1, 28, 469, 1, 28, 1, 1, 153

Offset: 1

Author

Ilya Gutkovskiy, Aug 21 2021

Keywords

• nonn
• easy

Examples

			a(18) = 28 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 cubes of 1 and 3 then add them i.e., a(18) = 1^3 + 3^3 = 28. - _David A. Corneth_, Feb 24 2024
		

Links

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

Crossrefs

Cf. A001158, A051000, A069288, A069289, A280375, A347162, A347173, A347175.

Programs

Mathematica

Table[DivisorSum[n, #^3 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(2 k - 1)^3 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sum(k=0, sqrtint(n), if ((k%2) && !(n%k), k^3)); \\ 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]^3
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

Formula

G.f.: Sum_{k>=1} (2*k - 1)^3 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).