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A125180 a(n) = 2*a(n-1) + prime(n) - prime(n-1), a(1)=2, where prime(n) denotes the n-th prime.

%I A125180 #37 Jul 31 2025 16:45:38
%S A125180 2,5,12,26,56,114,232,466,936,1878,3758,7522,15048,30098,60200,120406,
%T A125180 240818,481638,963282,1926568,3853138,7706282,15412568,30825142,
%U A125180 61650292,123300588,246601178,493202360,986404722,1972809448,3945618910
%N A125180 a(n) = 2*a(n-1) + prime(n) - prime(n-1), a(1)=2, where prime(n) denotes the n-th prime.
%C A125180 Row sums of A125179.
%C A125180 Limit_{n->oo} a(n)/a(n-1) = 2.
%H A125180 James C. McMahon, <a href="/A125180/b125180.txt">Table of n, a(n) for n = 1..1000</a>
%F A125180 a(n) = prime(n) + Sum_{j=1..n-1} 2^(n-j-1)*prime(j), where prime(k) denotes the k-th prime.
%F A125180 a(n) = Sum_{i=0..prime(n)-1} 2^(n-1-pi(i)), where prime(n) = A000040(n) and pi(n) = A000720(n). - _Ridouane Oudra_, Oct 17 2019
%F A125180 a(1) = 2 and a(n) = prime(n) + Sum_{i=1..n-1} a(i) for n > 1. - _Alexandre Herrera_, Dec 10 2023
%F A125180 a(n) = A000040(n) + A110299(n-1), for n > 1. - _Ridouane Oudra_, Jul 27 2025
%e A125180 a(4)=26 because 4*prime(1)+2*prime(2)+prime(3)+prime(4) = 8+6+5+7 = 26.
%p A125180 a[1]:=2: for n from 2 to 35 do a[n]:=2*a[n-1]+ithprime(n)-ithprime(n-1) od: seq(a[n],n=1..35);
%t A125180 a[1] = 2; a[n_] := 2*a[n - 1] + Prime[n] - Prime[n - 1]; Table[a[n], {n, 1, 31}] (* _James C. McMahon_, Dec 10 2023 *)
%o A125180 (Magma) [n eq 1 select 2 else 2*Self(n-1)+NthPrime(n)-NthPrime(n-1):n in [1..31]]; // _Marius A. Burtea_, Oct 17 2019
%Y A125180 Cf. A125179, A000040, A000720, A110299.
%K A125180 nonn
%O A125180 1,1
%A A125180 _Gary W. Adamson_, Nov 22 2006
%E A125180 Edited by _N. J. A. Sloane_, Dec 02 2006