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%I #38 Dec 20 2024 17:44:09
%S 35,247,1247,2501,4187,7957,15251,17767,33227,49051,81317,118301,
%T 128627,182527,241001,250717,265651,302177,318551,438751,485357,
%U 563347,655051,679057,736751,753667,886657,981317,1010651,1090987,1163801,1361837,1563151
%N Semiprimes of the form (4*n + 1)*(6*n + 1) = 24*n^2 + 10*n + 1.
%C The first few values of n such that both n and n+1 give semiprimes in the sequence begin: 2607, 4017, 4062, 5967, 7107, 8472, 8892, ... In such cases, numbers of the form 10n+8 can always be expressed as the sum of the two primes 4n+1 and 6n+7. - _Wesley Ivan Hurt_, Feb 27 2015
%F a(n) = A033570(A130800(n)) = A033570(2*A255607(n)). - _M. F. Hasler_, Dec 13 2019
%e 35 is in the sequence because 35 = 5*7 and 5, 7 are primes of the form 4*k+1 and 6*k+1 respectively.
%e 247 is in the sequence because 247 = 13*19: both 13, 19 are primes of the form 6*k+1 and 13 also has the form 4*k+1.
%t Select[Table[24 n^2 + 10 n + 1, {n, 300}], PrimeOmega[#] == 2 &] (* or *) f[n_] := Last /@ FactorInteger[n] == {1, 1}; Select[Array[24 #^2 + 10 # + 1 &, 300], f[#] &]
%o (Magma) [(4*n+1)*(6*n+1): n in [1..300] | IsPrime(4*n+1) and IsPrime(6*n+1)];
%o (Magma) IsSemiprime:=func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..300] | IsSemiprime(s) where s is 24*n^2+10*n+1];
%o (PARI) for(n=1,250,if(bigomega(s=24*n^2+10*n+1)==2,print1(s,", "))) \\ _Derek Orr_, Feb 28 2015
%Y Subsequence of A245365.
%Y Cf. A001358, A002144, A002476, A113941, A255607 (associated n).
%Y Equals A033570(A130800). - _M. F. Hasler_, Dec 13 2019
%K nonn
%O 1,1
%A _Vincenzo Librandi_, Feb 27 2015