cp's OEIS Frontend

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.

%I A284278 #44 Oct 20 2021 21:30:46
%S A284278 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 A284278 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 A284278 30,2,31,61,32,2,33,2,34,2,35,2,36,2,37,73,38,2,39
%N 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.
%C A284278 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 A284278 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 A284278 From the definition it easily follows that, for a positive x, the sequence contains roughly equal numbers of prime and composite terms <= x.
%C A284278 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 A284278 Michael De Vlieger, <a href="/A284278/b284278.txt">Table of n, a(n) for n = 1..10000</a>
%H A284278 Michael De Vlieger, <a href="/A284278/a284278.png">Log-log scatterplot of a(n)</a> for n=1..2^12.
%e A284278 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 A284278 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 A284278 a[1]:=2;
%t A284278 a[n_?EvenQ]:=n/2+2;
%t A284278 a[n_?OddQ]:=If[PrimeQ[(n+1)/2+1], n+2, 2];
%t A284278 Map[a, Range[150]] (* _Peter J. C. Moses_, Mar 24 2017 *)
%o A284278 (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 A284278 Cf. A284172.
%K A284278 nonn
%O A284278 1,1
%A A284278 _Vladimir Shevelev_, Mar 24 2017
%E A284278 More terms from _Peter J. C. Moses_, Mar 24 2017