A050259 -id:A050259 - cp's OEIS frontend

A001915 Primes p such that the congruence 2^x == 3 (mod p) is solvable.

2, 5, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 83, 97, 101, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 239, 263, 269, 293, 307, 311, 313, 317, 347, 349, 359, 373, 379, 383, 389, 409, 419, 421, 431, 443, 461, 467, 479, 491, 499, 503, 509, 523

Offset: 1

N. J. A. Sloane

The sequence is known to be infinite [Polya] - thanks to Pieter Moree and Daniel Stefankovic for this comment, Dec 21 2009.

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 63.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Xianwen Wang, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
G. Polya, Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen, J. reine und angewandte Mathematik (Crelle), Volume 1921, Issue 151, Pages 1-31.

Cf. A001916, A050259, A123988.

N:= 1000: # to search the first N primes
{2} union select(t -> numtheory[mlog](3,2,p) <> FAIL, {seq(ithprime(n),n=2..N)});
# Robert Israel, Feb 15 2013

Select[Prime[Range[120]], MemberQ[Table[Mod[2^x-3, #], {x, 0, #}], 0]&] (* Jean-François Alcover, Aug 29 2011 *)
Monitor[aaa=Reap[Do[p=Prime[m];sol=MultiplicativeOrder[2,p,{3}];If[IntegerQ[sol],Sow[p]],{m,1000}]],{m}];tmp=Transpose[{1+Range[Length[aaa[[2,1]]]],aaa[[2,1]]}] (* Xianwen Wang, Jul 22 2025 *)

isok(p) = isprime(p) && sum(k=0, (p-1), Mod(2, p)^k == 3); \\ Michel Marcus, Mar 12 2017

is(n)=isprime(n) && (n==2 || #znlog(3, Mod(2, n))) \\ Charles R Greathouse IV, Aug 15 2018

Better description from Joe K. Crump (joecr(AT)carolina.rr.com), Dec 11 2000

More terms from David W. Wilson, Dec 12 2000

A277554 Positive integers n such that 7^n == 3 (mod n).

Original entry on oeis.org

1, 2, 46, 2227, 6684830083, 12827743861, 151652531182, 155657642297, 3102126273955, 11006109076099, 50473807426174, 172794904196354

Offset: 1

Max Alekseyev, Oct 19 2016

No other terms below 10^15.

Cf. A066438, A277126.
Cf. Solutions to 7^n == k (mod n): A277371 (k=-3), A277370 (k=-2), A015954 (k=-1), A067947 (k=1), A277401 (k=2).
Cf. Solutions to b^n == 3 (mod n): A050259 (b=2), A130422 (b=4), A123061 (b=5), A116629 (b=13).

isok(n) = Mod(7, n)^n == 3; \\ Michel Marcus, Oct 20 2016

A296370 Numbers m such that 2^m == 3/2 (mod m).

Original entry on oeis.org

1, 111481, 465793, 79036177, 1781269903307, 250369632905747, 708229497085909, 15673900819204067

Offset: 1

Max Alekseyev, Dec 11 2017

Equivalently, 2^(m+1) == 3 (mod m).
Also, numbers m such that 2^(m+1) - 2 is a Fermat pseudoprime base 2, i.e., 2^(m+1) - 2 belongs to A015919 and A006935.
Some larger terms (may be not in order): 2338990834231272653581, 341569682872976768698011746141903924998969680637.

OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): this sequence (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Select[Range[10^6], Divisible[2^(# + 1) - 3, #] &] (* Robert Price, Oct 11 2018 *)

a(n) = A296104(n) - 1.

A123988 Primes p such that 2^x == 3 (mod p) has no solutions.

Original entry on oeis.org

3, 7, 17, 31, 41, 43, 73, 79, 89, 103, 109, 113, 127, 137, 151, 157, 199, 223, 229, 233, 241, 251, 257, 271, 277, 281, 283, 331, 337, 353, 367, 397, 401, 433, 439, 449, 457, 463, 487, 521, 569, 571, 593, 601, 607, 617, 631, 641, 673, 683, 691, 727, 733, 739, 751, 761, 809, 811, 823, 857, 881, 911

Offset: 1

Artur Jasinski, Nov 23 2006

nonn

Such primes cannot divide solutions to 2^m == 3 (mod m) (see A050259).

J. K. Crump, Number Theory Web.

Cf. A050259, A001915 (complement in the primes).

lst:=[3]; for p in [5..911 by 2] do if IsPrime(p) then t:=0; e:=Ceiling(Log(2, p+1)); for x in [e..p-2] do if 2^x mod p eq 3 then t:=1; break; end if; end for; if t eq 0 then Append(~lst, p); end if; end if; end for; lst; // Arkadiusz Wesolowski, Jan 12 2021

Edited by Max Alekseyev, Jan 14 2007
Corrected by Max Alekseyev, Jun 08 2011
Corrected by Arkadiusz Wesolowski, Jan 12 2021

A296104 Numbers k such that 2^k == 3 (mod k-1).

Original entry on oeis.org

2, 111482, 465794, 79036178, 1781269903308, 250369632905748, 708229497085910, 15673900819204068

Offset: 1

Krzysztof Ziemak and Max Alekseyev, Dec 04 2017

Also, numbers k such that 2^k - 2 is a Fermat pseudoprime, i.e., 2^k - 2 belongs to A015919 and A006935.
a(3) was found by McDaniel (1989).
Some larger terms (maybe not in order): 2338990834231272653582, 341569682872976768698011746141903924998969680638.
Discovered huge even PSP(2) numbers of the form 2*M(n), where n=p*q and M(n)=2^n-1, ensure that the following numbers are also even pseudoprimes of the form 2*M(p)*M(q): 2*M(37)*M(12589), 2*M(131)*M(17854891864360859951), 2*M(179)*M(1398713032993), 2*M(2111)*M(335494787819), 2*M(35267)*M(50508121). - Krzysztof Ziemak, Jan 01 2018

W. L. McDaniel, Some Pseudoprimes and Related Numbers Having Special Forms, Math. Comp. 53:187 (1989), 407-409.

Cf. A000079, A015919, A006935, A014741, A050259, A055685, A296370.

k = 2; lst = {2}; While[k < 1000000001, If[ PowerMod[2, k, k -1] == 3, AppendTo[lst, k]]; k += 10; If[ PowerMod[2, k, k -1] == 3, AppendTo[lst, k]]; k += 2]; lst (* Robert G. Wilson v, Jan 01 2018 *)

is_A296104(n) = Mod(2, n-1)^n == 3; \\ Iain Fox, Dec 07 2017

A296104_list = [n for n in range(2,10**6) if pow(2,n,n-1) == 3 % (n-1)] # Chai Wah Wu, Dec 06 2017

a(n) = A296370(n) + 1.

A127437 Duplicate of A001915.

Original entry on oeis.org

2, 5, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 83, 97, 101, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 239, 263, 269, 293, 307, 311, 313, 317, 347, 349, 359, 373, 379, 383, 389, 409, 419, 421, 431, 443, 461, 467, 479, 491, 499, 503, 509, 523, 541, 547, 557, 563, 577

Offset: 1

Max Alekseyev, Jan 14 2007

dead

Potential prime divisors of solutions to 2^m == 3 (mod m) (see A050259).

Minimal nonnegative solutions to 2^x == 3 (mod a(n)) are given in A127438.

Cf. A050259, A123988 (complement in the primes).

forprime(p=5,1000, g=znprimroot(p); u=znlog(Mod(2,p),g); v=znlog(Mod(3,p),g); if( v%u==0, print1(p,", "); ))

Corrected by Max Alekseyev, Jun 08 2011

Corrected by Arkadiusz Wesolowski, Jan 12 2021

A334634 Numbers m that divide 2^m + 11.

Original entry on oeis.org

1, 13, 16043199041, 91118493923, 28047837698634913

Offset: 1

Max Alekseyev, Sep 10 2020

Equivalently, numbers m such that 2^m == -11 (mod m).

No other terms below 10^17.

OEIS Wiki, 2^n mod n

Solutions to 2^n == k (mod n): A296370 (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), this sequence (k=-11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12).

a(5) from Sergey Paramonov, Oct 10 2021

cp's OEIS Frontend

A001915 Primes p such that the congruence 2^x == 3 (mod p) is solvable.

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A277554 Positive integers n such that 7^n == 3 (mod n).

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A296370 Numbers m such that 2^m == 3/2 (mod m).

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A123988 Primes p such that 2^x == 3 (mod p) has no solutions.

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A296104 Numbers k such that 2^k == 3 (mod k-1).

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A127437 Duplicate of A001915.

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A334634 Numbers m that divide 2^m + 11.

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