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a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * prime(k).
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%I #15 Jan 30 2025 03:53:09

%S 2,3,10,7,20,23,58,19,44,51,112,63,140,151,328,53,114,117,250,131,276,

%T 287,604,161,342,355,742,383,798,825,1720,131,270,273,566,289,596,607,

%U 1252,323,664,675,1392,711,1458,1481,3046,407,832,839,1718,875,1782

%N a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * prime(k).

%H Robert Israel, <a href="/A333176/b333176.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{k=1..n} (-1)^A010060(n-k) * (binomial(n,k) mod 2) * a(k) = prime(n).

%p N:= 200: # for a(1) .. a(N)

%p P:= [seq(ithprime(i),i=1..N)]:

%p B:= [1,1]: R:= 2:

%p for n from 2 to N do

%p B:= [1,op(B[2..-1]+B[1..-2] mod 2),1];

%p R:= R, convert(P[select(t -> B[t+1] = 1,[$1..n])],`+`);

%p od:

%p R; # _Robert Israel_, Jan 29 2025

%t Table[Sum[Mod[Binomial[n, k], 2] Prime[k], {k, 1, n}], {n, 1, 53}]

%o (PARI) a(n) = sum(k=1, n, if (binomial(n, k) % 2, prime(k))); \\ _Michel Marcus_, Mar 10 2020

%Y Cf. A000040, A007443, A007504, A010060, A030015, A050513.

%K nonn,look,changed

%O 1,1

%A _Ilya Gutkovskiy_, Mar 10 2020