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

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

1,9

Links

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A231484 A340398 A347172 * A352032 A051001 A363991

Adjacent sequences: A347172 A347173 A347174 * A347176 A347177 A347178

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Aug 21 2021

Status

approved