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A347174
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Sum of cubes of odd divisors of n that are <= sqrt(n).
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3
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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
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OFFSET
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1,9
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} (2*k - 1)^3 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).
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EXAMPLE
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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
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MATHEMATICA
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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