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%I #25 Mar 23 2024 21:06:15
%S 1,2,3,5,6,4,5,7,8,6,7,11,12,8,9,10,11,13,14,12,13,17,18,14,15,16,17,
%T 19,20,18,19,23,24,20,21,22,23,29,30,24,25,26,27,28,29,31,32,30,31,37,
%U 38,32,33,34,35,36,37,41,42,38,39,40,41,43,44,42,43,47
%N a(n) depends on the primality of a(n-1) and parity of n (see Comments lines for definition).
%C Initial term a(1)=1 and thereafter:
%C If n is even and a(n-1) is prime, then a(n) = least prime > a(n-1).
%C If n is even and a(n-1) is composite and a(n-2) is prime, then a(n) = 1 + greatest prime < a(n-2).
%C Otherwise, a(n) = a(n-1) + 1.
%C Every integer will be seen, with many seen twice. Largest first differences will be at prime gaps. Starting at 3, pairs of consecutive primes appear. Interestingly, the values seen once seem to correspond to A093513 except for 2.
%e From 2 we move to 3, it is prime, so go to 5. Next evaluation to 6, having departed from a prime, so go to 3 + 1 = 4. Next eval move to 5, it is prime, so go to 7. Next eval to 8, having departed from a prime, so go to 5 + 1 = 6. Next eval move to 7, it is prime, so go to 11. Next eval move to 12, having departed from a prime, so go to 7 + 1 = 8. Next eval move to 9. Next eval move to 10. Next eval move to 11, it is prime, so go to 13. This example adds the terms 3, 5, 6, 4, 5, 7, 8, 6, 7, 11, 12, 8, 9, 10, 11, 13.
%t a[1] = 1; a[n_] := a[n] = If[EvenQ[n], If[PrimeQ[a[n-1]], NextPrime[a[n-1] + 1], If[CompositeQ[a[n-1]] && PrimeQ[a[n-2]], 1 + NextPrime[a[n-2], -1], a[n-1] + 1]], a[n-1] + 1]; Array[a, 100] (* _Amiram Eldar_, Mar 19 2024 *)
%Y Cf. A093513.
%K nonn,easy
%O 1,2
%A _Bill McEachen_, Mar 13 2024
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