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A173937 Smallest natural d = d(n) such that 2^n + d is lesser of twin primes (n = 0, 1, 2, ...).

%I A173937 #18 Jun 16 2025 15:18:45
%S A173937 2,1,1,3,1,9,7,9,13,9,7,33,31,27,67,33,1,39,7,63,313,105,277,9,73,69,
%T A173937 457,51,121,105,7,219,91,297,247,321,115,567,1327,411,553,987,325,183,
%U A173937 2065,2565,415,879,241,459,643,1209,391,1155,1477,1449,175,129,1045
%N A173937 Smallest natural d = d(n) such that 2^n + d is lesser of twin primes (n = 0, 1, 2, ...).
%D A173937 Richard K. Guy, Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994
%D A173937 Friedhelm Padberg, Elementare Zahlentheorie, Spektrum Akademischer Verlag, Berlin Heidelberg, 1996
%D A173937 Daniel Shanks, Solved and Unsolved Problems in Number Theory, 4th ed., New York, Chelsea, 1993
%H A173937 Robert G. Wilson v, <a href="/A173937/b173937.txt">Table of n, a(n) for n = 0..3000</a> (0..300 from T. D. Noe).
%H A173937 Ken Takusagawa, <a href="http://www.mit.edu/~kenta/three/prime/a.html">Twin primes</a>
%e A173937 2^0 + 2 = 3 = prime(2), 2^0 + 4 = 5 = prime(3).
%e A173937 2^1 + 1 = 3 = prime(2), 2^1 + 3 = 5 = prime(3).
%e A173937 2^2 + 1 = 5 = prime(3), 2^2 + 3 = 7 = prime(4).
%t A173937 Join[{2}, Table[s = 2^n + 1; While[! (PrimeQ[s] && PrimeQ[s + 2]), s = s + 2]; s - 2^n, {n, 60}]] (* _T. D. Noe_, May 08 2012 *)
%o A173937 (PARI) A173937(n)={forstep(p=2^n\6*6+5,2<<n,6,isprime(p)||next; isprime(p+2)&return(p-2^n)); 2-n} \\ _M. F. Hasler_, Oct 21 2012
%Y A173937 Cf. A001359, A006512, A124001, A208572 (smallest twin prime > 2^n).
%K A173937 nonn
%O A173937 0,1
%A A173937 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 03 2010
%E A173937 Values a(0..300) double-checked by _M. F. Hasler_, Oct 21 2012