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A354973 a(0)=0; for n > 0, a(n) = 2*a(n-1) if n-1 is prime, a(n-1) + 1 otherwise.

%I A354973 #32 Jul 07 2022 02:23:41
%S A354973 0,1,2,4,8,9,18,19,38,39,40,41,82,83,166,167,168,169,338,339,678,679,
%T A354973 680,681,1362,1363,1364,1365,1366,1367,2734,2735,5470,5471,5472,5473,
%U A354973 5474,5475,10950,10951,10952,10953,21906,21907,43814,43815,43816,43817,87634
%N A354973 a(0)=0; for n > 0, a(n) = 2*a(n-1) if n-1 is prime, a(n-1) + 1 otherwise.
%F A354973 a(n) = A110299(k) - 2^k + n + 1, where k = primepi(n-1) and taking A110299(0) = 0. - _Kevin Ryde_, Jun 22 2022
%e A354973 5 is prime, so a(6) = 2*a(5) = 2*9 = 18.
%e A354973 6 is not prime, so a(7) = a(6) + 1 = 18 + 1 = 19.
%t A354973 a[0] = 0; a[n_] := a[n] = If[PrimeQ[n - 1], 2*a[n - 1], a[n - 1] + 1]; Array[a, 50, 0] (* _Amiram Eldar_, Jun 21 2022 *)
%o A354973 (Python)
%o A354973 from sympy import isprime
%o A354973 a = [0]; [a.append(2*a[-1] if isprime(n) else a[-1]+1) for n in range(48)]
%o A354973 print(a) # _Michael S. Branicky_, Jun 21 2022
%o A354973 (PARI) a(n) = my(k=primepi(n-1)); fromdigits(primes(k),2) - 1<<k + n + 1; \\ _Kevin Ryde_, Jun 22 2022
%Y A354973 Cf. A110299.
%K A354973 nonn
%O A354973 0,3
%A A354973 _Ben White_, Jun 14 2022