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A317929
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A235199, which is a multiplicative permutation of natural numbers.
4
1, 1, 3, 3, 7, 3, 5, 5, 27, 7, 17, 9, 13, 5, 21, 35, 11, 27, 19, 21, 15, 17, 23, 15, 147, 13, 135, 15, 43, 21, 59, 63, 51, 11, 35, 81, 37, 19, 39, 35, 41, 15, 29, 51, 189, 23, 73, 105, 75, 147, 33, 39, 53, 135, 119, 25, 57, 43, 31, 63, 61, 59, 135, 231, 91, 51, 67, 33, 69, 35, 107, 135, 47, 37, 441, 57, 85, 39
OFFSET
1,3
COMMENTS
Multiplicative because A235199 is.
Question: Are all terms positive? No negative terms in range 1 .. 2^18. Also checked up to n = 2^18 that the denominators match with A299150.
LINKS
FORMULA
a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A235199(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
PROG
(PARI)
up_to = 16384;
A235199(n) = if(n<=4, n, my(f = factor(n)); for(i=1, #f~, if(5==f[i, 1], f[i, 1] += 2, if(7==f[i, 1], f[i, 1] -= 2, my(k=primepi(f[i, 1])); if(k>4, f[i, 1] = prime(A235199(k)))))); factorback(f));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v317929aux = DirSqrt(vector(up_to, n, A235199(n)));
A317929(n) = numerator(v317929aux[n]);
CROSSREFS
Cf. A235199, A299150 (seems to give the denominators).
Cf. also A317930.
Sequence in context: A096915 A249806 A249382 * A285387 A100803 A036840
KEYWORD
nonn,frac,mult
AUTHOR
Antti Karttunen, Aug 23 2018
STATUS
approved