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URL: https://oeis.org/A317831

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A317831
Numerators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).
8
1, 3, 2, 19, 3, 3, 4, 63, 9, 9, 6, 19, 7, 6, 6, 867, 9, 27, 10, 57, 8, 9, 12, 63, 11, 21, 11, 19, 15, 9, 16, 3069, 12, 27, 12, 171, 19, 15, 14, 189, 21, 12, 22, 57, 27, 18, 24, 867, 41, 33, 18, 133, 27, 33, 18, 63, 20, 45, 30, 57, 31, 24, 18, 22199, 21, 18, 34, 171, 24, 18, 36, 567, 37, 57, 22, 95, 24, 21, 40, 2601, 227, 63, 42, 19, 27, 33
OFFSET
1,2
FORMULA
a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A000203(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
Sum_{k=1..n} A317831(k) / A317832(k) ~ n^2 * sqrt(Pi/(24*log(n))) * (1 - (gamma - 1 + 6*zeta'(2)/Pi^2) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025
PROG
(PARI)
A317831perA317832(n) = if(1==n, n, (sigma(n)-sumdiv(n, d, if((d>1)&&(d<n), A317831perA317832(d)*A317831perA317832(n/d), 0)))/2);
A317831(n) = numerator(A317831perA317832(n));
(PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/((1-p*X)*(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025
CROSSREFS
Cf. A317832 (gives the denominators).
Cf. also A000203, A299151.
Sequence in context: A006281 A157705 A185447 * A078073 A317927 A075568
KEYWORD
nonn,frac
AUTHOR
Antti Karttunen, Aug 10 2018
STATUS
approved