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A214029 The union of the disjoint prime sequences A000057 and A106535. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000057, A001177, A106535.

Sequence in context: A038937 A092940 A045326 * A195602 A105897 A129386

Adjacent sequences:  A214026 A214027 A214028 * A214030 A214031 A214032

KEYWORD

nonn

AUTHOR

Art DuPre, Jul 09 2012

STATUS

approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)