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

1,2

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

Formula

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

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v317930aux = DirSqrt(vector(up_to, n, A234840(n)));
A317930(n) = numerator(v317930aux[n]);

Crossrefs

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

Cf. also A317929.

Sequence in context: A170924 A173007 A113099 * A270078 A269875 A271203

Adjacent sequences: A317927 A317928 A317929 * A317931 A317932 A317933

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 23 2018

Status

approved