A229064 Lesser of Fermi-Dirac twin primes: both a(n)(>=5) and a(n)+2 are in A050376.

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

Offset: 1

Author

Vladimir Shevelev, Sep 17 2013

Keywords

• nonn

Comments

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

Examples

			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.
		

References

• V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000
• S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Crossrefs

Cf. A001359.

Programs

Mathematica

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 *)

PARI

isok(n)={my(e1=isprimepower(n), e2=isprimepower(n+2)); n >= 5 && e1 && e2 && !bitand(e1,e1-1) && !bitand(e2,e2-1)} \\ Andrew Howroyd, Oct 16 2024