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A110299 a(n) = Sum_{i=0..n-1} 2^i*prime(n-i).

%I A110299 #33 Feb 17 2024 12:47:44
%S A110299 2,7,19,45,101,215,447,913,1849,3727,7485,15007,30055,60153,120353,
%T A110299 240759,481577,963215,1926497,3853065,7706203,15412485,30825053,
%U A110299 61650195,123300487,246601075,493202253,986404613,1972809335,3945618783,7891237693,15782475517
%N A110299 a(n) = Sum_{i=0..n-1} 2^i*prime(n-i).
%D A110299 Eric Angelini, "Array with primes." Pers. comm. on the SeqFan mailing list, Sep. 7 2005.
%H A110299 Alois P. Heinz, <a href="/A110299/b110299.txt">Table of n, a(n) for n = 1..1000</a>
%F A110299 G.f: b(x)/(1-2*x), where b(x) is the g.f. of A000040. - _Mario C. Enriquez_, Dec 10 2016
%F A110299 a(n) = 2*a(n-1) + A000040(n) for n>0 with a(0)=0. - _Alois P. Heinz_, Dec 10 2016
%F A110299 From _Ridouane Oudra_, Jan 25 2024: (Start)
%F A110299 a(n) = Sum_{i=0..prime(n+1)-1} (2^(n-pi(i)) - 1), where prime(n) = A000040(n) and pi(n) = A000720(n).
%F A110299 a(n) = A125180(n+1) - A000040(n+1);
%F A110299 a(n) = Sum_{i=1..n} A125180(i);
%F A110299 a(n) = A007504(n) + Sum_{i=1..n-1} a(i). (End)
%p A110299 a:= proc(n) option remember;
%p A110299       `if`(n=0, 0, ithprime(n)+2*a(n-1))
%p A110299     end:
%p A110299 seq(a(n), n=1..30);  # _Alois P. Heinz_, Dec 10 2016
%t A110299 Table[Sum[2^i * Prime[n-i], {i, 0, n-1}], {n, 1, 30}]
%o A110299 (PARI) a(n) = fromdigits(primes(n),2); \\ _Kevin Ryde_, Jun 22 2022
%o A110299 (Magma)
%o A110299 A110299:= func< n | (&+[2^(n-j)*NthPrime(j): j in [1..n]]) >;
%o A110299 [A110299(n): n in [1..40]]; // _G. C. Greubel_, Jan 03 2023
%o A110299 (SageMath)
%o A110299 @CachedFunction # a = A110299
%o A110299 def a(n): return 2 if (n==1) else 2*a(n-1) + nth_prime(n)
%o A110299 [a(n) for n in range(1,41)] # _G. C. Greubel_, Jan 03 2023
%o A110299 (Python)
%o A110299 from sympy import prime
%o A110299 def A110299(n):
%o A110299     c = 0
%o A110299     for i in range(n):
%o A110299         c = (c<<1)+prime(i+1)
%o A110299     return c # _Chai Wah Wu_, Jan 04 2023
%Y A110299 Cf. A000040, A135483, A287353.
%Y A110299 Cf. A125180, A000720, A007504
%K A110299 nonn,easy
%O A110299 1,1
%A A110299 _Ryan Propper_, Sep 07 2005