A317930 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A234840, which is a multiplicative permutation of natural numbers.

1, 3, 1, 27, 19, 3, 61, 135, 3, 57, 11, 27, 281, 183, 19, 2835, 101, 9, 5, 513, 61, 33, 263, 135, 1083, 843, 5, 1647, 29, 57, 59, 15309, 11, 303, 1159, 81, 1811, 15, 281, 2565, 1091, 183, 157, 297, 57, 789, 409, 2835, 11163, 3249, 101, 7587, 541, 15, 209, 8235, 5, 87, 31, 513, 7, 177, 183, 168399, 5339, 33, 1013, 2727

Offset: 1

Author

Antti Karttunen, Aug 23 2018

Keywords

• nonn
• frac
• mult

Comments

Multiplicative because A234840 is.

Question: Are all terms positive? No negative terms in range 1 .. 2^17. Also (checked for n <= 2^17) the denominators seem to be given by A317932.

Links

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

Crossrefs

Cf. A234840, A317932 (seems to give denominators, see A261179).

Cf. also A317929.

Programs

PARI

up_to = 16384;
A234840(n) = if(n<=1,n,my(f = factor(n)); for(i=1, #f~, if(2==f[i,1], f[i,1]++, if(3==f[i,1], f[i,1]--, f[i,1] = prime(-1+A234840(1+primepi(f[i,1])))))); factorback(f)); \\ Antti Karttunen, Aug 23 2018
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.
v317930aux = DirSqrt(vector(up_to, n, A234840(n)));
A317930(n) = numerator(v317930aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A234840(n) - Sum_{d|n, d>1, d 1.