A267758 If a(n) is prime, then a(n) = a(n+1) - a(n-1): lexicographic first permutation of the nonnegative integers with this property.

0, 1, 2, 3, 5, 8, 4, 6, 7, 13, 20, 9, 10, 11, 21, 12, 14, 15, 16, 17, 33, 18, 19, 37, 56, 22, 23, 45, 24, 25, 26, 27, 28, 29, 57, 30, 31, 61, 92, 32, 34, 35, 36, 38, 39, 40, 41, 81, 42, 43, 85, 44, 46, 47, 93, 48, 49, 50, 51, 52, 53, 105, 54, 55, 58, 59

Offset: 0

Author

Eric Angelini and M. F. Hasler, Jan 21 2016

Keywords

• nonn

Comments

In practice, the definition implies that if a prime term a(n) has been added, the next term is computed as a(n+1) = a(n) + a(n-1). Will it ever happen that a term computed that ("greedy") way would be equal to a number which already occurred previously? This is of course forbidden, so the term a(n) which was "tentatively" added has to be changed, as to be either composite, or to "produce" a "legal" a(n+1). In case a(n) itself was already computed as sum of the two preceding terms (because of a prime a(n-1)), this backtracking must go one step further. But since the sum of two primes > 2 is even, there will never be three primes > 2 in a row.

As a result (independently found by Lars Blomberg, private communication), there cannot be more than the three lines that can already be seen in the graph of a(0..1000): (i) a(n) ~ n if the predecessor is composite, (ii) a(n) ~ 2n if a(n-1) is prime but a(n-2) isn't, (iii) a(n) ~ 3n if a(n-1) and a(n-2) both are prime. - M. F. Hasler, Jan 25 2016

Lars Blomberg also observed that (i) the sequence of the lesser of two consecutive odd primes is (at least up to n = 1000) the same as A217199, and (ii) the "isolated primes" are always (up to n = 10^7) preceded by an even composite number. We can show that the latter implies that a(n+1) = a(n) + a(n-1) (with prime a(n)) will never be equal to a previously used number, which in that case must lie on the "upper line" of the (even) a(m+1) = a(m-2) + 2*a(m-1) ~ 3m, m < n, in order to occur before a(n+1) ~ 2n, so that we would have a sequence "odd, prime, even". - M. F. Hasler, Jan 25 2016

Links

• M. F. Hasler, Table of n, a(n) for n = 0..1000
• Eric Angelini, Prime a(n) as difference of its neighbours, SeqFan list, Jan. 21, 2016.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A217199.

Programs

PARI

{N=200; a=[]; U=[]; L=0; while(#a
			
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