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

Author

Antti Karttunen, Aug 23 2018

Keywords

• nonn
• frac
• mult

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

Crossrefs

Cf. A235199, A299150 (seems to give the denominators).

Cf. also A317930.

Programs

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&&dA317937.
v317929aux = DirSqrt(vector(up_to, n, A235199(n)));
A317929(n) = numerator(v317929aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A235199(n) - Sum_{d|n, d>1, d 1.