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%I #45 Jul 07 2014 00:30:18
%S 5,7,9,11,17,23,29,41,47,59,71,79,81,101,107,137,149,167,179,191,197,
%T 227,239,269,281,311,347,359,419,431,461,521,569,599,617,641,659,809,
%U 821,827,839,857,881,1019,1031,1049,1061,1091,1151,1229,1277,1289,1301,1319,1367
%N Lesser of Fermi-Dirac twin primes: both a(n)(>=5) and a(n)+2 are in A050376.
%C Terms of A050376 play the role of primes in Fermi-Dirac arithmetic. Therefore, if q and q+2 are consecutive terms of A050376, then we call them twin primes in Fermi-Dirac arithmetic. The sequence lists lessers of them.
%C There exist conjecturally only 5 Fermat primes F, such that both F-1 and F are in A050376. If we add pair (3,4), then we obtain exactly 6 such pairs as an analog of the unique pair (2,3) in usual arithmetic, which is not considered as a pair of twin primes.
%C For n>4, numbers n such that n and n+2 are of the form p^(2^k), where p is prime and k >= 0. - _Ralf Stephan_, Sep 23 2013
%C If a(n) is not the lesser of twin primes (A001359), then either a(n) or a(n)+2 is a perfect square. For example, a(4)=9 and a(7)=23. Note that the first case is possible only if a(n) = 3^(2^m), m>=1. - _Vladimir Shevelev_, Jun 27 2014
%D V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].
%H Peter J. C. Moses, <a href="/A229064/b229064.txt">Table of n, a(n) for n = 1..10000</a>
%H S. Litsyn and V. S. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%e 2, 3 are not in the sequence, although pairs (2,4) and (3,5) are in A050376. Indeed, 2 and 4 as well as 3 and 5 are not consecutive terms of A050376.
%t inA050376Q[1]:=False; inA050376Q[n_] := Length[#] == 1 && (Union[Rest[IntegerDigits[#[[1]][[2]], 2]]] == {0} || #[[1]][[2]] == 1)&[FactorInteger[n]]; nextA050376[n_] := NestWhile[#+1&, n+1, !inA050376Q[#] == True&]; Select[Range[1500], inA050376Q[#] && (nextA050376[#]-#) == 2&] (* _Peter J. C. Moses_, Sep 19 2013 *)
%o (PARI) is(n)=if(n<5,return(false);m=factor(n);mm=factor(n+2);e=m[1,2];ee=mm[1,2];matsize(m)[1]==1&&matsize(mm)[1]==1&&e==2^valuation(e,2)&&ee=2^valuation(ee,2) /* _Ralf Stephan_, Sep 22 2013 */
%Y Cf. A001359.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Sep 17 2013
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