A284278 - OEIS

A284278

a(1)=2; for n >= 1, if n+2 is prime then a(2*n+1) = 3*n + 2 and a(2*n) = n + 2, otherwise all terms are 2.

2, 3, 5, 4, 2, 5, 9, 6, 2, 7, 13, 8, 2, 9, 2, 10, 2, 11, 21, 12, 2, 13, 25, 14, 2, 15, 2, 16, 2, 17, 33, 18, 2, 19, 37, 20, 2, 21, 2, 22, 2, 23, 45, 24, 2, 25, 2, 26, 2, 27, 2, 28, 2, 29, 57, 30, 2, 31, 61, 32, 2, 33, 2, 34, 2, 35, 2, 36, 2, 37, 73, 38, 2, 39

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OFFSET

1,1

COMMENTS

The sequence is motivated by A284172, by the message from B. Jubin dated Mar 23 2017 and by the classic open problem of showing that there are infinitely many primes p for which 2*p-1 is also prime. If there were only finitely many such primes, then there would be a place where this sequence is generated by the same rule as A284172.

The sequence of the first differences begins 1, 2, -1, -2, 3, 4, -3, -4, 5, 6, -5, -6, 7, -7, 8, -8, 9, 10, -9, -10, 11, 12, -11, -12, 13, -13, 14, -14, 15, 16, -15, -16, 17, 18, ...

From the definition it easily follows that, for a positive x, the sequence contains roughly equal numbers of prime and composite terms <= x.

A conditional property: if there is a maximal prime P such that 2*P-1 is also prime, then for n > P, every pair (a(2*n), a(2*n+1)) contains one prime and one composite. Indeed, if n+2 is prime, then a(2*n) = n + 2 is prime, while a(2*n+1) = 2*n + 3 = 2*(n+2) - 1 is composite; if n+2 is composite, then a(2*n) = n + 2 is composite, while a(2*n+1) = 2 is prime. - Vladimir Shevelev, Mar 26 2017

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log-log scatterplot of a(n) for n=1..2^12.

EXAMPLE

For n=19, a(38) = a(2*19) = 19+2 = 21, a(39) = a(2*19+1) = 2, the latter since 19+2 is not prime;

for n=21, a(42) = a(2*21) = 21+2 = 23, a(43) = a(2*21+1) = 2*21+3 = 45 since 21+2 is prime.

MATHEMATICA

a[1]:=2;

a[n_?EvenQ]:=n/2+2;

a[n_?OddQ]:=If[PrimeQ[(n+1)/2+1], n+2, 2];

Map[a, Range[150]] (* Peter J. C. Moses, Mar 24 2017 *)

PROG