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.

1, 3, 4, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839

Offset: 1

Author

Alzhekeyev Ascar M, May 19 2011

Keywords

• nonn
• more

Comments

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.

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

Any exceptions to the conjecture are larger than 10^5. - Charles R Greathouse IV, Oct 03 2022

Examples

			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.
		

Crossrefs

A174265 is a subsequence.

Cf. A144671, A000043.

Programs

Maple

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:
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

Mathematica

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

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

Extensions

a(20)-a(23) from Jean-François Alcover, Oct 30 2019

a(24)-a(28) from Charles R Greathouse IV, Sep 16 2022

a(29) from Charles R Greathouse IV, Sep 29 2022

a(30)-a(33) from Bill McEachen, Jul 30 2024

Definition simplified by Max Alekseyev, Dec 04 2024