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A237662 Primes of the form 2^(k+l+m+1) - 2^(l+m+1) + 2^(m+1) + l - 2 1
3, 7, 11, 17, 23, 31, 37, 47, 59, 67, 73, 101, 127, 131, 191, 223, 229, 239, 251, 257, 383, 401, 457, 479, 503, 521, 577, 991, 997, 1019, 1031, 1153, 1601, 1993, 2039, 2053, 2069, 3583, 3593, 3851, 3967, 4079, 4091, 4099, 4111, 4133, 6143, 6211 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

k can be zero, with the arbitrary limitation, when k = 0, l <= m.

When k = m = 1 we get primes of the form 2^(l + 2) + l + 2.

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.

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).

LINKS

Table of n, a(n) for n=1..48.

EXAMPLE

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.

PROG

(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], ", "))

CROSSREFS

Cf. A235860, A236547, A007931, A129962, A237816

Sequence in context: A289750 A184856 A023234 * A134707 A047838 A188653

Adjacent sequences:  A237659 A237660 A237661 * A237663 A237664 A237665

KEYWORD

nonn

AUTHOR

Robin Garcia, Feb 11 2014

STATUS

approved

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Last modified September 22 17:35 EDT 2019. Contains 327311 sequences. (Running on oeis4.)