A284278 - OEIS

%I #44 Oct 20 2021 21:30:46

%S 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,

%T 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,

%U 30,2,31,61,32,2,33,2,34,2,35,2,36,2,37,73,38,2,39

%N 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.

%C 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.

%C 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, ...

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

%C 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

%H Michael De Vlieger, <a href="/A284278/b284278.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A284278/a284278.png">Log-log scatterplot of a(n)</a> for n=1..2^12.

%e 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;

%e 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.

%t a[1]:=2;

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

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

%t Map[a, Range[150]] (* _Peter J. C. Moses_, Mar 24 2017 *)

%o (PARI) a(n) = if(n<2, 2, if(n%2, if(isprime((n + 1)/2 + 1), n + 2, 2), (n/2 + 2))); \\ _Indranil Ghosh_, Mar 25 2017

%Y Cf. A284172.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Mar 24 2017

%E More terms from _Peter J. C. Moses_, Mar 24 2017