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 A221211 Numbers n such that n and n + 4 are prime and there is a power of two in the interval (n,n+4). 7
 3, 7, 13, 127 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It is a conjecture that this is a finite sequence. These may be the only known cousin primes with this property. The Cf.s list similar sequences, of the form -- numbers n such that n and n+m are prime and contain a power of two in the interval (n,n+m). The case where m=2, the twin prime case -- not listed, has only one member n=3. Another member would have to be a twin where n+2 was a Fermat type prime and n a Mersenne prime. LINKS PROG (MAGMA) //Program finds primes separated by an even number (called gap) which //have a power of two between them. The program starts with the smallest //of two above gap. Primes less than this starting point can be checked by //inspection. In this example 3 also works. gap:=4; start:=Ilog2(gap)+1; for i:= start to 1000 do     powerof2:=2^i;     for k:=powerof2-gap+1 to powerof2-1 by 2 do         if (IsPrime(k) and IsPrime(k+gap)) then k;         end if;     end for; end for; (MAGMA) [n: n in PrimesUpTo(10^3) | IsPrime(n+4) and exists{t: t in [n+1..n+3 by 2] | IsOne(t/2^Valuation(t, 2))}]; // Bruno Berselli, May 16 2013 CROSSREFS Cf. A023200. Cf. A220951 (gap of 6), A213210 (8), A220746 (10), A213677 (12), A222424 (14), A222227 (16), A222219 (18). Sequence in context: A038691 A237890 A082718 * A322301 A154821 A084741 Adjacent sequences:  A221208 A221209 A221210 * A221212 A221213 A221214 KEYWORD nonn AUTHOR Brad Clardy, Feb 21 2013 STATUS approved

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)