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

24, 36, 100, 112, 120, 240, 288, 320, 372, 472, 532, 576, 600, 810, 828, 864, 882, 924, 990, 1088, 1104, 1164, 1180, 1208, 1236, 1284, 1360, 1392, 1482, 1508, 1560, 1584, 1620, 1632, 1692, 1740, 1818, 1900, 1920, 1938

Offset: 1

Author

Zak Seidov, Jun 20 2017

Keywords

• nonn

Comments

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}.

Examples

			24 is a term because 24 = 11+13 and 24 = 10+14.
Alternatively, 24 = A001043(5) = A118717(4), 36 = A001043(7) = A118717(6).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A001043 and A118717.

Programs

Mathematica

sp=Select[Range[4,1000],2==PrimeOmega[#]&]; Select[Table[sp[[k]]+sp[[k+1]],{k,100}], #==(p=NextPrime[#/2,-1])+NextPrime[p]&]
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 *)

PARI

issemi(n)=bigomega(n)==2
nextsp(x)=x=ceil(x); while(!issemi(x), x++); x
has(n)=precprime((n-1)/2)+nextprime(n/2)==n
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