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A284102 Numbers that are the sum of 10 consecutive primes and also the sum of 10 consecutive semiprimes.

%I A284102 #16 Mar 21 2017 13:38:20
%S A284102 6504,12946,12990,19052,19764,21490,31638,35604,41300,42364,45212,
%T A284102 52528,58104,60034,63400,66662,67858,69880,74090,74824,78542,88844,
%U A284102 96256,96346,97818,104584,106970,111122,113120,117540,125384
%N A284102 Numbers that are the sum of 10 consecutive primes and also the sum of 10 consecutive semiprimes.
%H A284102 Charles R Greathouse IV, <a href="/A284102/b284102.txt">Table of n, a(n) for n = 1..10000</a>
%e A284102 a(1)=6504 because 6504 is the sum of 10 consecutive primes A000040(114..114+9)={619,631,641,643,647,653,659,661,673,677} and also
%e A284102 6504 is the sum of 10 consecutive semiprimes A001358(192..192+9)={629,633,634,635,649,655,662,667,669,671}.
%e A284102 Note that a(1) = 6504 = A283873(10).
%p A284102 N:= 10^6:
%p A284102 P:= select(isprime, [$1..N]):
%p A284102 S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]):
%p A284102 P10:= {seq(add(P[i],i=m..m+9),m=1..nops(P)-9)}:
%p A284102 S10:= {seq(add(S[i],i=m..m+9),m=1..nops(S)-9)}:
%p A284102 sort(convert(P10 intersect S10,list)); # _Robert Israel_, Mar 20 2017
%t A284102 With[{nn = 12600}, Intersection[Total /@ Partition[Prime@ Range@ PrimePi@ nn, 10, 1], Total /@ Partition[Select[Range@ nn, PrimeOmega@ # == 2 &], 10, 1]]] (* _Michael De Vlieger_, Mar 20 2017 *)
%o A284102 (PARI) list(lim)=if(lim<6504,return([])); my(v=List(),u=v,P=primes(9),x=(lim+10*log(lim))\1,t); forprime(p=2,x\2, forprime(q=2,min(x\p,p), listput(u,p*q))); u=Set(u); while(u[#u]+1+(t=sum(i=0,8,u[#u-i]))<=lim, for(n=x+1,lim-t, if(issemi(n), u=concat(u,n); next(2))); break); for(i=1,#u-9, u[i]+=sum(j=1,9,u[i+j])); t=vecsum(P); forprime(p=P[#P]+1,, t+=p; if(t>lim, break); if(setsearch(u,t), listput(v,t)); t-=P[1]; P=concat(P[2..9], p)); Vec(v) \\ _Charles R Greathouse IV_, Mar 20 2017
%Y A284102 Cf. A000040, A001358, A127337, A283873.
%K A284102 nonn
%O A284102 1,1
%A A284102 _Zak Seidov_, Mar 20 2017