A317929 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A235199, which is a multiplicative permutation of natural numbers.

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

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Adjacent sequences: A317926 A317927 A317928 * A317930 A317931 A317932

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 23 2018

Status

approved