A347175 - OEIS

%I #14 Feb 24 2024 11:03:40

%S 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,

%T 707,1,1,82,1,626,82,1,1,82,626,1,82,1,1,707,1,1,82,2402,626,82,1,1,

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

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

%H David A. Corneth, <a href="/A347175/b347175.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 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

%t Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 75}]

%t 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

%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]^4

%o 		,

%o 			return(res)

%o 		)

%o 	); res

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

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

%K nonn,easy

%O 1,9

%A _Ilya Gutkovskiy_, Aug 21 2021