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

1,1

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

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.

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

Cf. A001359.

Sequence in context: A028885 A228233 A256091 * A024910 A283771 A045236

Adjacent sequences: A229061 A229062 A229063 * A229065 A229066 A229067

Keyword

nonn

Author

Vladimir Shevelev, Sep 17 2013

Status

approved