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A190213 Integers m such that m divides (2^m-2)^2 and (m-2)^((k-1)*(1+k*(m-1))) == 1 (mod k), where k = 2^m - 1.

%I A190213 #46 Feb 14 2025 02:55:17
%S A190213 1,3,4,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,
%T A190213 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,
%U A190213 216091,756839
%N A190213 Integers m such that m divides (2^m-2)^2 and (m-2)^((k-1)*(1+k*(m-1))) == 1 (mod k), where k = 2^m - 1.
%C A190213 Original definition: let k=2^n-1 and m=1+(k-1)*(n-1), x=m*k and define remainders a and b via 2^(x-1) == (a+1) (mod x) and m^(x-1) == (b+1) (mod x). If a == 0 (mod k) and b == 0 (mod k), n is in the sequence.
%C A190213 Conjecture: All odd entries are also Mersenne exponents (A000043): primes n such that 2^n-1 is prime.
%C A190213 Any exceptions to the conjecture are larger than 10^5. - _Charles R Greathouse IV_, Oct 03 2022
%e A190213 For n=3, k=2^3-1=7, m=1+6*2=13, x=m*k=13*7=91, 2^(x-1)==(a+1) (mod x) with 2^90 == (63+1)(mod 91), fixes a=63. m^(x-1) == (b+1) (mod x) with 13^90 == (77+1) (mod 91) fixes b=77. The two conditions are satisfied: 63 == 0 (mod 7) and 77 == 0 (mod 7). Therefore n=3 is in the sequence.
%p A190213 isA190213 := proc(n) local k,m,x,a,b ; k := 2^n-1 ; m := (k-1)*(n-1)+1 ; x := k*m ; a := modp( 2 &^ (x-1),x) -1 ; b := modp( m &^ (x-1),x) -1 ; return ( modp(a,k) = 0 and modp(b,k)=0 ) ; end proc:
%p A190213 for n from 2 do if isA190213(n) then printf("%d,\n",n); end if; end do; # avoids n=1 and undefined 0^0, _R. J. Mathar_, Jun 11 2011
%t A190213 okQ[n_] := Module[{k, m, x, a, b}, k = 2^n - 1; m = 1 + (k - 1)(n - 1); x = m k; a = PowerMod[2, x - 1, x] - 1; b = PowerMod[m, x - 1, x] - 1; Mod[a, k] == 0 && Mod[b, k] == 0];
%t A190213 Reap[For[n = 1, n < 10^4, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Oct 30 2019 *)
%o A190213 (PARI) is(n)=my(k=2^n-1,m=(k-1)*(n-1)+1,e=m*k-1); Mod(2,k)^e==1 && Mod(m,k)^e==1 \\ _Charles R Greathouse IV_, Sep 16 2022
%Y A190213 A174265 is a subsequence.
%Y A190213 Cf. A144671, A000043.
%K A190213 nonn,more
%O A190213 1,2
%A A190213 _Alzhekeyev Ascar M_, May 19 2011
%E A190213 a(20)-a(23) from _Jean-François Alcover_, Oct 30 2019
%E A190213 a(24)-a(28) from _Charles R Greathouse IV_, Sep 16 2022
%E A190213 a(29) from _Charles R Greathouse IV_, Sep 29 2022
%E A190213 a(30)-a(33) from _Bill McEachen_, Jul 30 2024
%E A190213 Definition simplified by _Max Alekseyev_, Dec 04 2024