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A138290 Numbers m such that 2^(m+1) - 2^k - 1 is composite for all 0 <= k < m.

%I A138290 #47 Mar 23 2024 20:01:39
%S A138290 6,14,22,26,30,36,38,42,54,57,62,70,78,81,90,94,110,122,126,132,134,
%T A138290 138,142,147,150,158,166,168,171,172,174,178,182,190,194,198,206,210,
%U A138290 222,238,254,285,294,312,315,318,334,336,350,366,372,382,405,414,416,432
%N A138290 Numbers m such that 2^(m+1) - 2^k - 1 is composite for all 0 <= k < m.
%C A138290 The binary representation of 2^(m+1) - 2^k - 1 has m 1-bits and one 0-bit. Note that prime m are very rare: 577 is the first and 5569 is the second.
%C A138290 A208083(a(n)+1) = 0 (cf. A081118). - _Reinhard Zumkeller_, Feb 23 2012 [Corrected by _Thomas Ordowski_, Feb 19 2024]
%C A138290 Conjecture: 2^j - 2 are terms for j > 2. - _Chai Wah Wu_, Sep 07 2021
%C A138290 The proof of this conjecture is in A369375. - _Thomas Ordowski_, Mar 20 2024
%H A138290 Chai Wah Wu, <a href="/A138290/b138290.txt">Table of n, a(n) for n = 1..996</a> (terms 1..275 from T. D. Noe)
%F A138290 For these m, A095058(m) = 0 and A110700(m) > 1.
%F A138290 For n > 0, a(n) = A369375(n+1) - 1. - _Thomas Ordowski_, Mar 20 2024
%e A138290 6 is here because 95, 111, 119, 123, 125 and 126 are all composite.
%t A138290 t={}; Do[num=2^(n+1)-1; k=0; While[k<n && !PrimeQ[num-2^k], k++ ]; If[k==n, AppendTo[t,n]], {n,100}]; t
%t A138290 Select[Range[500],AllTrue[2^(#+1)-1-2^Range[0,#-1],CompositeQ]&] (* _Harvey P. Dale_, Apr 09 2022 *)
%o A138290 (Haskell)
%o A138290 import Data.List (elemIndices)
%o A138290 a138290 n = a138290_list !! (n-1)
%o A138290 a138290_list = map (+ 1) $ tail $ elemIndices 0 a208083_list
%o A138290 -- _Reinhard Zumkeller_, Feb 23 2012
%o A138290 (Python)
%o A138290 from sympy import isprime
%o A138290 A138290_list = []
%o A138290 for n in range(1,10**3):
%o A138290     k2, n2 = 1, 2**(n+1)
%o A138290     for k in range(n):
%o A138290         if isprime(n2-k2-1):
%o A138290                 break
%o A138290         k2 *= 2
%o A138290     else:
%o A138290         A138290_list.append(n) # _Chai Wah Wu_, Sep 07 2021
%o A138290 (PARI) isok(m) = my(nb=0); for (k=0, m-1, if (!ispseudoprime(2^(m+1) - 2^k - 1), nb++, break)); nb==m; \\ _Michel Marcus_, Sep 13 2021
%Y A138290 Many common terms with A092112.
%Y A138290 Cf. A081118, A095058, A110700, A208083, A369375.
%K A138290 nonn
%O A138290 1,1
%A A138290 _T. D. Noe_, Mar 13 2008