A108974 Sort the primes (except 2) according to the multiplicative order of 2 modulo that prime. If two primes have the same order of 2, they are arranged numerically.

3, 7, 5, 31, 127, 17, 73, 11, 23, 89, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 178481, 241, 601, 1801, 2731, 262657, 29, 113, 233, 1103, 2089, 331, 2147483647, 65537, 599479, 43691, 71, 122921, 37, 109, 223, 616318177, 174763, 79

Offset: 1

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Jul 27 2005

Keywords

• nonn

Comments

Or, primitive prime divisors of the Mersenne numbers 2^n-1 (see A000225) in their order of occurrence.

Of course the Mersenne primes 2^p-1 (cf. A000043) appear in this sequence.

If all odd positive numbers, not just the odd primes, are sorted in this way, the result is A059912. - Jeppe Stig Nielsen, Feb 13 2020

Examples

			The order of 2 modulo 3 is 2 and the order of 2 modulo 7 is 3. So 3 comes before 7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..4275
• G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
• Jeppe Stig Nielsen, A108974 arranged as an irregular array.
• K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Crossrefs

Cf. A000043, A000225, A001348, A014664, A059912, A086251.

Programs

Mathematica

a = 1; DeleteDuplicates[Flatten[#[[All, 1]] & /@ FactorInteger[Table[a = 2 a + 1, {i, 1, 30}]]]] (* Horst H. Manninger, Mar 20 2021 *)

PARI

do(n)=my(v=List(),P=1,g,t,f); for(k=2,n, t=2^k-1; g=P; while((g=gcd(g,t))>1, t/=g); f=factor(t)[,1]; for(i=1,#f, listput(v,f[i])); P*=t); Vec(v) \\ Charles R Greathouse IV, Sep 23 2016

Extensions

More terms from Martin Fuller, Sep 25 2006