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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.

%I A108974 #34 Mar 28 2021 01:10:05
%S A108974 3,7,5,31,127,17,73,11,23,89,13,8191,43,151,257,131071,19,524287,41,
%T A108974 337,683,47,178481,241,601,1801,2731,262657,29,113,233,1103,2089,331,
%U A108974 2147483647,65537,599479,43691,71,122921,37,109,223,616318177,174763,79
%N 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.
%C A108974 Or, primitive prime divisors of the Mersenne numbers 2^n-1 (see A000225) in their order of occurrence.
%C A108974 Of course the Mersenne primes 2^p-1 (cf. A000043) appear in this sequence.
%C A108974 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
%H A108974 Charles R Greathouse IV, <a href="/A108974/b108974.txt">Table of n, a(n) for n = 1..4275</a>
%H A108974 G. Everest et al., <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
%H A108974 Jeppe Stig Nielsen, <a href="/A108974/a108974.txt">A108974 arranged as an irregular array</a>.
%H A108974 K. Zsigmondy, <a href="https://doi.org/10.1007%2FBF01692444">Zur Theorie der Potenzreste</a>, Monatsh. Math., 3 (1892), 265-284.
%e A108974 The order of 2 modulo 3 is 2 and the order of 2 modulo 7 is 3. So 3 comes before 7.
%t A108974 a = 1; DeleteDuplicates[Flatten[#[[All, 1]] & /@ FactorInteger[Table[a = 2 a + 1, {i, 1, 30}]]]] (* _Horst H. Manninger_, Mar 20 2021 *)
%o A108974 (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
%Y A108974 Cf. A000043, A000225, A001348, A014664, A059912, A086251.
%K A108974 nonn
%O A108974 1,1
%A A108974 Douglas Stones (dssto1(AT)student.monash.edu.au), Jul 27 2005
%E A108974 More terms from _Martin Fuller_, Sep 25 2006