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