A347173 -id:A347173 - cp's OEIS frontend

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

Ilya Gutkovskiy, Aug 21 2021

			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

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

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

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

a(n) = sum(k=0, sqrtint(n), if ((k%2) && !(n%k), k^3)); \\ Michel Marcus, Aug 22 2021

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

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

A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 1, 626, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 2402, 626, 82, 1, 1, 82, 626, 2402, 82, 1, 1, 707, 1, 1, 2483, 1, 626, 82, 1, 1, 82, 3027, 1, 82, 1, 1, 707

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

			a(18) = 82 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 sum of fourth powers of 1 and 3 then add them i.e., a(18) = 1^4 + 3^4 = 82. - _David A. Corneth_, Feb 24 2024

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

Cf. A001159, A051001, A069288, A069289, A347143, A347172, A347173, A347174.

Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(2 k - 1)^4 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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]^4
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

G.f.: Sum_{k>=1} (2*k - 1)^4 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).

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

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A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).

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