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

%I A347175 #14 Feb 24 2024 11:03:40
%S A347175 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 A347175 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 A347175 82,626,2402,82,1,1,707,1,1,2483,1,626,82,1,1,82,3027,1,82,1,1,707
%N A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).
%H A347175 David A. Corneth, <a href="/A347175/b347175.txt">Table of n, a(n) for n = 1..10000</a>
%F A347175 G.f.: Sum_{k>=1} (2*k - 1)^4 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).
%e A347175 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 A347175 Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 75}]
%t A347175 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 A347175 (PARI)
%o A347175 a(n) = {
%o A347175 	my(s = sqrtint(n), res);
%o A347175 	n>>=valuation(n, 2);
%o A347175 	d = divisors(n);
%o A347175 	for(i = 1, #d,
%o A347175 		if(d[i] <= s,
%o A347175 			res += d[i]^4
%o A347175 		,
%o A347175 			return(res)
%o A347175 		)
%o A347175 	); res
%o A347175 } \\ _David A. Corneth_, Feb 24 2024
%Y A347175 Cf. A001159, A051001, A069288, A069289, A347143, A347172, A347173, A347174.
%K A347175 nonn,easy
%O A347175 1,9
%A A347175 _Ilya Gutkovskiy_, Aug 21 2021