A361259 - OEIS

%I #15 Mar 13 2023 11:52:02

%S 10,26,39,358,58,77,155,129,583,562,323,326,551,381,629,501,707,1294,

%T 789,791,961,1354,1159,1262,1369,1371,1591,1718,1849,1851,2271,2127,

%U 3561,2427,3077,2747,3085,3442,4811,3826,3829,3831,5089,4227,4659,4661,5345,7318,5587,8146,6333,6081,6338

%N a(n) is the least semiprime that is the sum of n consecutive primes.

%C No sum of two consecutive primes is a semiprime.  Proof: if prime(k) and prime(k+1) are odd primes, prime(k)+prime(k+1) is even, so the only way it could be a semiprime is for (prime(k)+prime(k+1))/2 to be prime.  But this is  impossible because prime(k) and prime(k+1) are consecutive primes and (prime(k)+prime(k+1))/2 is between them.

%H Robert Israel, <a href="/A361259/b361259.txt">Table of n, a(n) for n = 3..10000</a>

%e a(3) = 10 = 2+3+5, a(4) = 26 = 3+5+7+11, a(5) = 39 = 3+5+7+11+13, a(6) = 358 = 47+53+59+61+67+71.

%p P:= select(isprime, [2,seq(i,i=3..10^6,2)]):

%p S:= ListTools:-PartialSums(P):

%p f:= proc(n) local k,s;

%p   if numtheory:-bigomega(S[n])=2 then return S[n] fi;

%p   if n::even then

%p     for k from 1 do

%p       if isprime((S[n+k]-S[k])/2) then return S[n+k]-S[k] fi

%p     od

%p   else

%p     for k from 1 do

%p       if numtheory:-bigomega(S[n+k]-S[k]) = 2 then return S[n+k]-S[k] fi

%p     od

%p   fi

%p end proc:

%p map(f, [$3..100]);

%t pr = Prime[Range[10^6]]; Do[to = Total /@ Partition[pr,n,1]; se =

%t Select[to,2 == PrimeOmega[#] &,1][[1]]; AppendTo[s,se],{n,3,30}]; s

%Y Cf. A001358.

%K nonn

%O 3,1

%A _Zak Seidov_ and _Robert Israel_, Mar 06 2023