A190213 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.

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

1,2

Comments

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

Links

Table of n, a(n) for n=1..33.

Example

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.

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 *)

Prog

(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

Crossrefs

A174265 is a subsequence.

Cf. A144671, A000043.

Sequence in context: A360688 A079463 A325422 * A216574 A216561 A192269

Adjacent sequences: A190210 A190211 A190212 * A190214 A190215 A190216

Keyword

nonn,more

Author

Alzhekeyev Ascar M, May 19 2011

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

Status

approved