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Lexicographically earliest increasing sequence of semiprimes such that a(n) + a(n+1) is a semiprime, with a(0) = 4.
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%I #7 Jan 12 2024 22:46:46

%S 4,6,9,25,26,39,46,49,57,58,65,69,74,85,93,94,111,115,122,143,146,155,

%T 159,187,194,201,202,205,206,209,213,214,237,265,278,287,299,323,326,

%U 329,358,365,381,382,403,415,427,451,454,469,482,497,501,502,505,537,542,559,562,573,581,586,591,611

%N Lexicographically earliest increasing sequence of semiprimes such that a(n) + a(n+1) is a semiprime, with a(0) = 4.

%C The only case where two successive terms are even is 4, 6 at the start, since if 2*p and 2*q are semiprimes where p and q are odd primes, 2*p + 2*q is divisible by 4.

%H Robert Israel, <a href="/A368648/b368648.txt">Table of n, a(n) for n = 0..10000</a>

%e a(2) = 6 because 6 = 2 * 3 and 4 + 6 = 10 = 2 * 5 are semiprimes.

%e a(3) = 9 because 9 = 3 * 3 and 6 + 9 = 15 = 3 * 5 are semiprimes.

%e a(4) = 25 because 25 = 5 * 5 and 9 + 25 = 34 = 2 * 17 are semiprimes.

%p R:= 4: x:= 4: count:= 1:

%p for y from 5 do

%p if numtheory:-bigomega(y) = 2 and numtheory:-bigomega(x+y) = 2 then

%p R:= R,y; x:= y; count:= count+1;

%p if count = 100 then break fi

%p fi

%p od:

%p R;

%Y Cf. A001358, A361611.

%K nonn

%O 0,1

%A _Zak Seidov_ and _Robert Israel_, Jan 02 2024