A282028 If n is prime then a(n) = 2n, otherwise a(n) is the smallest missing number.

0, 1, 4, 6, 2, 10, 3, 14, 5, 7, 8, 22, 9, 26, 11, 12, 13, 34, 15, 38, 16, 17, 18, 46, 19, 20, 21, 23, 24, 58, 25, 62, 27, 28, 29, 30, 31, 74, 32, 33, 35, 82, 36, 86, 37, 39, 40, 94, 41, 42, 43, 44, 45, 106, 47, 48, 49, 50, 51, 118, 52, 122, 53, 54, 55, 56, 57, 134, 59, 60, 61, 142, 63, 146, 64, 65

Offset: 0

Author

N. J. A. Sloane, Feb 16 2017

Keywords

• nonn

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000720, A127202, A143545, A280985, A282029.

Programs

PARI

first(n) = { my(res = vector(n), i = 1); for(x=1, n-1, if(isprime(x), res[x+1] = 2*x, if(setsearch(Set(res), i), i++); res[x+1]=i; i++)); res; } \\ Iain Fox, Nov 18 2017

Formula

If n is prime, a(n) = 2n, and these points line on the upper straight line in the graph.

If n is not a prime, after n terms we have seen all the numbers from 0 through a(n) and also the doubles of all the primes p in the range a(n)/2 < p < n.

So n = a(n) + pi(n) - pi(a(n)/2). In other words, if n is not a prime then a(n) is the unique solution to a(n) - pi(a(n)/2) = n - pi(n).

This implies that if n is not a prime, a(n) = n*(1 - 1/(2*log(n)) + o(1/log(n))).

These are the points on the lower line, which is not straight but has slope roughly equal to 1.