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A229064
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Lesser of Fermi-Dirac twin primes: both a(n)(>=5) and a(n)+2 are in A050376.
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3
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5, 7, 9, 11, 17, 23, 29, 41, 47, 59, 71, 79, 81, 101, 107, 137, 149, 167, 179, 191, 197, 227, 239, 269, 281, 311, 347, 359, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 839, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1367
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OFFSET
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1,1
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COMMENTS
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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.
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.
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
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
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REFERENCES
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V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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