OFFSET

1,1

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

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.

EXAMPLE

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 *)

PROG

(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 */

CROSSREFS

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Sep 17 2013

STATUS

approved