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A214033
Places n where A214030(n) = n or A214030(n) = n-2.
0
13, 17, 19, 23, 37, 41, 47, 67, 89, 109, 137, 139, 157, 181, 191, 211, 229, 233, 239, 257, 277, 281, 283, 307, 311, 331, 349, 353, 359, 373, 379, 397, 479, 499, 503, 521, 523, 547, 571, 593, 599, 613, 617, 619, 641
OFFSET
1,1
COMMENTS
It always has been one of the great mysteries of mathematics, that the superdiagonal sequence of A001177 consists of prime numbers A000057.
Here, regarding A214031 and A214032,there is the further conjecture that these two disjoint sequences are primes and roughly comparable in density. It isn't clear that these two sequences have a density, without appealing to the Riemann Hypothesis, but they are certainly close to one another in growing size.
Since these two sequences are disjoint, it is natural to take their union.
PROG
(PARI)
{b23(n)=local(t, m=1, s=[n]); if (n<2, 0, while(1,
if(m%2, s=concat(s, 2), s=concat(s, 3));
t=contfracpnqn(concat(s, n));
t=contfrac(n*t[1, 1]/t[2, 1]);
if(t[1]<n^2||t[#t]<n^2, m++, break)); m)};
To print the sequence a(n) to the screen,
for(i=1, 500, if(b23(i)==i||b23(i)==i-2,
print1(i, ", ")));
CROSSREFS
KEYWORD
nonn
AUTHOR
Art DuPre, Jul 12 2012
STATUS
approved