%I #42 Dec 04 2024 09:47:26

%S 1,3,4,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,

%T 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,

%U 216091,756839

%N 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 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 Conjecture: All odd entries are also Mersenne exponents (A000043): primes n such that 2^n-1 is prime.

%C Any exceptions to the conjecture are larger than 10^5. - _Charles R Greathouse IV_, Oct 03 2022

%e 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 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 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 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 Reap[For[n = 1, n < 10^4, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Oct 30 2019 *)

%o (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 A174265 is a subsequence.

%Y Cf. A144671, A000043.

%K nonn,more,changed

%O 1,2

%A _Alzhekeyev Ascar M_, May 19 2011

%E a(20)-a(23) from _Jean-François Alcover_, Oct 30 2019

%E a(24)-a(28) from _Charles R Greathouse IV_, Sep 16 2022

%E a(29) from _Charles R Greathouse IV_, Sep 29 2022

%E a(30)-a(33) from _Bill McEachen_, Jul 30 2024

%E Definition simplified by _Max Alekseyev_, Dec 04 2024