|
|
A317930
|
|
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A234840, which is a multiplicative permutation of natural numbers.
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,frac,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|