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A288949 Numbers that are both the sum of two consecutive primes and the sum of two consecutive semiprimes.

%I A288949 #16 Jul 31 2023 17:50:09
%S A288949 24,36,100,112,120,240,288,320,372,472,532,576,600,810,828,864,882,
%T A288949 924,990,1088,1104,1164,1180,1208,1236,1284,1360,1392,1482,1508,1560,
%U A288949 1584,1620,1632,1692,1740,1818,1900,1920,1938
%N A288949 Numbers that are both the sum of two consecutive primes and the sum of two consecutive semiprimes.
%C A288949 Positions of a(n) in A001043 and A118717: {5, 4}, {7, 6}, {15, 17}, {16, 19}, {17, 21}, {30, 39}, {34, 48}, {37, 53}, {42, 60}, {51, 77}.
%H A288949 Charles R Greathouse IV, <a href="/A288949/b288949.txt">Table of n, a(n) for n = 1..10000</a>
%e A288949 24 is a term because 24 = 11+13 and 24 = 10+14.
%e A288949 Alternatively, 24 = A001043(5) = A118717(4), 36 = A001043(7) = A118717(6).
%t A288949 sp=Select[Range[4,1000],2==PrimeOmega[#]&]; Select[Table[sp[[k]]+sp[[k+1]],{k,100}], #==(p=NextPrime[#/2,-1])+NextPrime[p]&]
%t A288949 Module[{nn=2000,sp},sp=Total/@Partition[Select[Range[nn],PrimeOmega[#]==2&],2,1];Intersection[ sp,Total/@Partition[Prime[Range[nn]],2,1]]] (* _Harvey P. Dale_, Jul 31 2023 *)
%o A288949 (PARI) issemi(n)=bigomega(n)==2
%o A288949 nextsp(x)=x=ceil(x); while(!issemi(x), x++); x
%o A288949 has(n)=precprime((n-1)/2)+nextprime(n/2)==n
%o A288949 list(lim)=my(v=List(),last=4,t); forfactored(n=6,nextsp(lim\2), if(vecsum(n[2][,2])==2, if(has(t=last+n[1]) && t<=lim, listput(v,t)); last=n[1])); Vec(v) \\ _Charles R Greathouse IV_, Feb 19 2018
%Y A288949 Intersection of A001043 and A118717.
%K A288949 nonn
%O A288949 1,1
%A A288949 _Zak Seidov_, Jun 20 2017