A253178 -id:A253178

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A119591

Least k such that 2*n^k - 1 is prime.

+10
3

1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 2, 1, 10, 1, 1, 6, 1, 2, 6, 1, 2, 136, 1, 1, 6, 6, 1, 6, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 2, 4, 4, 1, 2, 1, 1, 44, 1, 1, 2, 1, 3, 2, 5, 3, 2, 2, 1, 4, 1, 768, 4, 1, 1, 52, 34, 2, 132, 1, 1, 14, 7, 1, 2, 2, 1, 8, 1, 2, 10, 1, 24, 60, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,4

COMMENTS

From Eric Chen, Jun 01 2015: (Start)

Conjecture: a(n) is defined for all n.

a(303) > 10000, a(304)..a(360) = {1, 2, 11, 1, 990, 1, 1, 2, 2, 4, 74, 5, 1, 10, 6, 6, 4, 1, 1, 2, 1, 9, 12, 1, 80, 2, 1, 1, 2, 14, 3, 2, 3, 1, 12, 1, 60, 36, 1, 8, 4, 34, 1, 522, 3, 15, 14, 1, 6, 2, 3, 1, 4, 5, 4, 10, 1}.

a(n) = 1 if and only if n is in A006254. (End)

From Eric Chen, Sep 16 2021: (Start)

Now a(303) is known to be 40174, also other terms > 10000: a(383) = 20956, a(515) = 58466, a(522) = 62288, a(578) = 129468, a(581) > 400000, a(590) = 15526, a(647) = 21576, a(662) = 16590, a(698) = 127558, a(704) = 62034, see the a-file and the references.

a(n) = 2 if and only if n is in A066049 but not in A006254.

a(n) = 3 if and only if n is in A214289 but not in A006254 or A066049. (End)

LINKS

Eric Chen, Table of n, a(n) for n = 2..580

Gary Barnes, Riesel conjectures and proofs

Eric Chen, Table of n, a(n) for n = 2..2050 status

Prime Wiki, Riesel prime small bases least n

FORMULA

From Eric Chen, Sep 16 2021: (Start)

a(6*n) = A098873(n).

a(2^n) = A279095(n).

a(A006254(n)) = 1.

a(A066049(n)) <= 2.

a(A214289(n)) <= 3. (End)

MATHEMATICA

f[n_] := Block[{k = 0}, While[ ! PrimeQ[2*n^k - 1], k++ ]; k ]; Table[f[n], {n, 2, 106}] (* Ray Chandler, Jun 08 2006 *)

PROG

(PARI) a(n) = for(k=1, 2^24, if(ispseudoprime(2*n^k-1), return(k))) \\ Eric Chen, Jun 01 2015

CROSSREFS

Cf. A119624, A253178.

Numbers r such that 2*k^r-1 is prime: A090748 (k=2), A003307 (k=3), A146768 (k=4), A120375 (k=5), A057472 (k=6), A002959 (k=7), ... (k=8), ... (k=9), A002957 (k=10), A120378 (k=11), ... (k=12), A174153 (k=13), A273517 (k=14), ... (k=15), ... (k=16), A193177 (k=17), A002958 (k=25).

KEYWORD

nonn,hard

AUTHOR

Pierre CAMI, Jun 01 2006

EXTENSIONS

Corrected and extended by Ray Chandler, Jun 08 2006

STATUS

approved

A119624

Least k>0 such that, for n>1, 2*n^k + 1 is prime; or 0 if no such prime possible as 2*n^k + 1 is 0 mod(3).

+10
3

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 1, 0, 47, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 2729, 1, 0, 1, 2, 0, 1, 2, 0, 175, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 3, 3, 0, 43, 1, 0, 1, 2, 0, 1, 1, 0, 3, 2, 0, 1, 1, 0, 3, 1, 0, 11, 1, 0, 1, 4, 0, 1, 2, 0, 1, 1, 0, 3, 2, 0, 1, 1, 0, 1, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

LINKS

Robert Israel, Table of n, a(n) for n = 1..217

MAPLE

f:= proc(n) local k;

if n mod 3 = 1 then return 0 fi;

if n mod 3 = 2 then r:= 2 else r:= 1 fi;

for k from 1 by r do if isprime(2*n^k+1) then return k fi od

end proc:

f(1):= 1:

map(f, [$1..100]); # Robert Israel, Apr 02 2018

MATHEMATICA

f[n_] := Block[{k = 0}, If[Mod[n, 3] != 1, k = 1; While[ ! PrimeQ[2*n^k + 1], k++ ]; ]; k ]; Table[f[n], {n, 2, 100}] (* Ray Chandler, Jun 08 2006 *)

Table[If[n>1 && Mod[n, 3]==1, 0, k=1; While[ !PrimeQ[2n^k+1], k++ ]; k], {n, 100}] (* T. D. Noe, Jun 08 2006 *)

PROG

(PARI) a(n) = if(n%3==1, 0, for(k=1, 2^24, if(ispseudoprime(2*n^k+1), return(k)))) \\ Eric Chen, Mar 20 2015

CROSSREFS

Cf. A119591, A253178.

KEYWORD

nonn

AUTHOR

Pierre CAMI, Jun 08 2006

EXTENSIONS

Extended by Ray Chandler and T. D. Noe, Jun 08 2006

STATUS

approved

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