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A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2.

%I A237662 #14 Jun 02 2025 09:21:03
%S A237662 3,7,11,17,23,31,37,47,59,67,73,101,127,131,191,223,229,239,251,257,
%T A237662 383,401,457,479,503,521,577,991,997,1019,1031,1153,1601,1993,2039,
%U A237662 2053,2069,3583,3593,3851,3967,4079,4091,4099,4111,4133,6143,6211
%N A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2.
%C A237662 These prime numbers can be written in the numeral system described in A235860 (perhaps not minimally) this way : 2..21..12..2 (or 1..12..2) k twos followed to the right by l ones, followed to the right by m twos.
%C A237662 k can be zero, with the arbitrary limitation, when k = 0, l <= m.
%C A237662 When k = m = 1 we get primes of the form 2^(l + 2) + l + 2.
%C A237662 It must be noted these primes include the Mersenne primes 3, 7, 31, 127, 8191, ... as well as the Fermat primes 3, 5, 17, 257, 65537, with the exception of 5. Mersenne primes can be represented by a one followed to the right by an even number of twos (if the number of twos is odd, the Mersenne number is divisible by 3) with the exception of 3 represented as 12. The Fermat numbers can be represented with three ones followed to the right by a Mersenne number of twos (2^t - 1 (t = 0, 1, 2, 3, 4, 5,...)) :  3 = 111 instead of shorter 12, 5 = 1112 instead of shorter  21, 17 = 111222, 257 = 1112222222, 65537 = 111222222222222222. The composite (divisible by 641) 2^32 + 1 : three ones followed to the right by thirty one twos. The second Fermat prime: 5, appears in this sequence if we let l > m and l <= 3 when k = 0.
%C A237662 By A235860 3, 7 , 17 and 31 can be represented as 12, 122, 111222, 12222 cases when k=0 (primes of the form 2^(m+1) + l - 2: 31 = 2^5 +1 -2). And 11, 73, 191 as 212, 211122, 2122222 (73 = 2^7 - 2^6 + 2^3 + 3 - 2).
%e A237662 For k=l=m=1, 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2 = 2^4 - 2^3 + 2^2 + 1 - 2 = 16 - 8 + 4 + 1 - 2 = 11, so 11 is in the sequence.
%o A237662 (PARI) n=10^5;e=89;a=1;if(a%2==0,a=a+1);b=ceil(log(n)/log(2));i=0;d=floor(b^(2.5));v=vector(d);for(n=0,b,for(p=a,b,if(n==0,x=p,x=b);forstep(m=a,x,2,c=2^(n+m+p+1)-2^(m+p+1)+2^(p+1)+m-2;if(isprime(c),i++;v[i]=c))));w=vecsort(v,,8);u=vector(#(w)-1);for(j=1,#(w)-1,u[j]=w[j+1]);if(e>#(u),e=#(u));s=vector(e);for(k=1,e,s[k]=u[k];print1(s[k], ", "))
%Y A237662 Cf. A235860, A236547, A007931, A129962, A237816
%K A237662 nonn
%O A237662 1,1
%A A237662 _Robin Garcia_, Feb 11 2014