A214029 - OEIS

A214029

The union of the disjoint prime sequences A000057 and A106535.

2, 3, 7, 11, 19, 23, 31, 43, 59, 67, 71, 79, 83, 103, 127, 131, 163, 167, 179, 191, 223, 227, 239, 251, 271, 283, 311, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 571, 587, 599, 607, 631, 643, 647, 659, 683, 719, 727

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OFFSET

1,1

COMMENTS

Just as A000057 can be generated by looking at the subscripts of the sequence A001177 which are one less than their values, A106535 can be generated by looking at the subscripts of the sequence A001177 which are one greater than their values.

It is a surprising fact that these two sequences A000057 and A106535 are disjoint. The also have approximately the same density, if these densities exist.

It would be interesting to be able to interpret the relation of this prime sequence to the entire set of Fibonacci sequences, i.e., those sequences satisfying f(n+2) = f(n+1) + f(n) with various initial conditions.

LINKS

Table of n, a(n) for n=1..55.

PROG

(PARI) {a(n, p) = local(t, m=1, s=[n]); if( n<2, 0, while( 1,

s=concat(s, p);

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)};

{p(m, n, p)=for(k=m, n, if(k-2==a(k, p)||k==a(k, p), print1(k, ”, “))); }

p(1, 800, 1);

CROSSREFS