A015919 -id:A015919 - cp's OEIS frontend

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

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.

A333269 Positive integers n such that 17^n == 2 (mod n).

Original entry on oeis.org

1, 3, 5, 3585, 4911, 5709, 1688565, 7361691, 16747709, 3226850283899, 8814126944005, 33226030397603

Offset: 1

Seiichi Manyama, Mar 14 2020

No other terms below 10^16. Some larger term: 95549099691107109423357503242294996525424418266995858732192019626694044445113. - Max Alekseyev, Jan 09 2025

Cf. Solutions to b^n == 2 (mod n): A015919 (b=2), A276671 (b=3), A130421 (b=4), A124246 (b=5), A277401 (b=7), A116622 (b=13), this sequence (b=17).

for(k=1, 1e6, if(Mod(17, k)^k==2, print1(k", ")))

A333269_list = [n for n in range(1,10**6) if n == 1 or pow(17,n,n) == 2] # Chai Wah Wu, Mar 14 2020

a(10)-a(12) from Max Alekseyev, Jan 09 2025

A067934 Let rep(k) = (10^k - 1)/9 be the k-th repunit number = 11111..1111 with k 1 digits, then sequence gives values of k such that rep(k) == 1 (mod k).

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 259

Offset: 1

Benoit Cloitre, Mar 05 2002

nonn

Due to Fermat's little theorem, all prime numbers except 3 are in the sequence. E.g., rep(17) = 1 + 17*653594771241830.
Numbers n such that 10^n == 10 (mod 9n). The number (10^n - 1)/9 is a term if and only if n is a term. - Thomas Ordowski, Apr 28 2018
Generally, the repunit theorem: Let integer b <> 1 and n be a positive integer. Define R_b(n) = (b^n-1)/(b-1) = N. Then R_b(N) == 1 (mod N) if and only if N == 1 (mod n). - Thomas Ordowski, Apr 28 2018
Proof: (b^N-1)/(b-1)-1 = (b^N-b)/(b-1) is divisible by N if and only if b^N-b is divisible by b^n-1. Since b^N-b == b^(N mod n)-b (mod b^n-1), we have that b^N-b is divisible by b^n-1 if and only if N == 1 (mod n). QED. - Max Alekseyev, Apr 28 2018
Terms which are not prime are 1 U A303608. - Robert G. Wilson v, Jun 13 2018
No multiples of 3 are in this sequence. - Eric Chen, Jun 13 2018
A005939 is subsequence. - Eric Chen, Jun 13 2018

			(10^11 - 1)/9 = 11111111111 == 1 (mod 11), so 11 is a term.
We also have the congruence 10^11 == 10 (mod 9*11).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000864, A015919, A303608, A005939.

{1}~Join~Select[Range[260], Mod[#2, #1] == 1 & @@ {#, (10^# - 1)/9} &] (* Michael De Vlieger, May 06 2018 *)
fQ[n_] := PowerMod[10, n, 9 n] == 10; fQ[1] = True; Select[Range@260, fQ] (* Robert G. Wilson v, Jun 13 2018 *)

is(n)=n==1 || ((10^n-1)/9)%n==1 \\ Eric Chen, Jun 13 2018

A216885 Primes p such that x^47 = 2 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Vincenzo Librandi, Sep 20 2012

Complement of A059257 relative to A000040.

a(n) = A015919(n+1) up to n=60, and then both sequences start to differ substantially. [Bruno Berselli, Sep 20 2012]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[p: p in PrimesUpTo(500) | exists(t){x: x in ResidueClassRing(p) | x^47 eq 2}];

ok[p_] := Reduce[Mod[x^47 - 2, p] == 0, x, Integers] =!= False; Select[Prime[Range[100]], ok]

A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 23, 31, 37, 41, 43, 53, 67, 71, 73, 83, 89, 97, 103, 149, 157, 199, 239, 251, 257, 271, 277, 293, 307, 313, 331, 337, 359, 383, 397, 421, 431, 433, 499, 541, 557, 571, 587, 599, 601, 613, 631, 653, 659, 661, 683, 691, 709, 727, 751, 769, 823, 887, 911, 983, 1009, 1021, 1031, 1049, 1051, 1063, 1129, 1163, 1217

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

Appears to be a subsequence of A015919, A045344, A052085, A064555 and A143578.

			5 = 12_3 and 12_9 = 11 are both prime, so 5 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235265, A235473 - A235479, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 3], 9] &] (* Giovanni Resta, Sep 12 2019 *)

is(p,b=9,c=3)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

A165285 Primes which are the sum of at least 2 consecutive pars "Prime and PreviousNumber".

Original entry on oeis.org

17, 43, 59, 73, 79, 101, 109, 139, 163, 191, 197, 233, 239, 283, 317, 331, 379, 419, 433, 439, 443, 463, 467, 499, 521, 569, 571, 599, 617, 619, 641, 739, 743, 787, 811, 863, 911, 919, 941, 967, 971, 1021, 1039, 1061, 1063, 1087, 1097, 1109, 1117, 1229, 1289

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 13 2009

(1+2)+(2+3)+(4+5)=17, (4+5)+(6+7)+(10+11)=43, (6+7)+(10+11)+(12+13)=59,..

Cf. A130097, A062090, A050150, A015919

lst={};Do[s=2*Prime[m]-1;Do[p=Prime[n];s+=(2*p-1);If[PrimeQ[s],If[s<=6793,AppendTo[lst,s]]],{n,m+1,3*5!}],{m,1,3*5!}];lst=Take[Union@lst,200]

A244149 a(n) = 2(ndenominator(((n-1)(n^2)+2^(n+1)-4)/(2n))-n)/n+1.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 7, 5, 9, 1, 11, 1, 13, 9, 15, 1, 17, 1, 19, 13, 21, 1, 23, 9, 25, 17, 3, 1, 29, 1, 31, 21, 33, 69, 35, 1, 37, 25, 39, 1, 41, 1, 43, 5, 45, 1, 47, 13, 49, 33, 51, 1, 53, 109, 55, 37, 57, 1, 59, 1, 61, 41, 63, 25, 65, 1, 67, 45, 9, 1, 71, 1, 73, 49, 75, 153, 77, 1, 79

Offset: 1

Juri-Stepan Gerasimov, Sep 01 2014

nonn

			a(1) = 2*(1*denominator(((1-1)*(1^2)+2^(1+1)-4)/(2*1))-1)/1+1 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000040, A001567, A015919, A065091.

[2*(n*Denominator(((n-1)*(n^2)+2^(n+1)-4)/(2*n))-n)/n+1: n in [1..100]];

A244149[n_] := 2*(n*Denominator[((n-1)*n^2 + 2^(n+1) - 4)/(2*n)] - n)/n + 1;
Array[A244149, 100] (* Paolo Xausa, Jan 27 2025 *)

a(n) = 2*(n*denominator(((n-1)*(n^2)+2^(n+1)-4)/(2*n))-n)/n + 1; \\ Michel Marcus, Sep 03 2014

A294646 a(n) = (1/2)^(2n) mod (2n+1).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 4, 1, 1, 16, 1, 11, 25, 1, 1, 25, 4, 1, 10, 1, 1, 16, 1, 36, 13, 1, 9, 43, 1, 1, 16, 61, 1, 52, 1, 1, 64, 60, 1, 79, 1, 16, 22, 1, 64, 70, 44, 1, 70, 1, 1, 16, 1, 1, 28, 1, 59, 16, 4, 67, 31, 11, 1, 97, 1, 106, 79, 1, 1, 106, 69, 136, 100, 1, 1, 52, 64, 1, 40, 32, 1, 31, 1, 131, 169

Offset: 1

Robert Israel and Thomas Ordowski, Nov 05 2017

nonn

a(n) is the smallest k > 0 such that k*2^(2*n) == 1 (mod 2*n+1).
a(n)*A177023(n) == 1 (mod 2*n+1).
a(n)=1 iff 2*n+1 is in A015919.
1 <= a(n) <= 2*n, and is always coprime to 2*n+1.
Conjecture: a(n) is never 2 or 2*n or 2*n-2.
a(n) = 2*n-1 iff 2*n+1 is in A006521.

			For n = 3, 2*n+1 = 7, (1/2)^6 == 4^6 == 1 (mod 7) so a(3)=1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006521, A015919, A177023.

seq((1/2 mod (2*n+1)) &^(2*n) mod (2*n+1), n=1..200);

a(n) = (1/2)^(2*n) % (2*n+1); \\ Michel Marcus, Nov 06 2017

A296117 Base-2 pseudoprimes of the form 2pq where p and q are primes.

Original entry on oeis.org

161038, 49699666, 760569694, 4338249646, 357647681422, 547551530002, 3299605275646, 22999986587854, 42820164121582, 55173914702146, 69345154539266, 353190859033982

Offset: 1

Max Alekseyev, Dec 05 2017

a(5) and a(10) are found by McDaniel (1989).

Terms in this sequence are of the form 2pq where p and q are distinct odd primes (A075819). - Charles R Greathouse IV, Dec 05 2017

W. L. McDaniel, Some pseudoprimes and related numbers having special forms, Math. Comp. 53:187 (1989), 407-409.

Subsequence of A006935 and hence of A015919.
The even terms of A215672.
Intersection of A006935 and A215672. - Felix Fröhlich, Dec 05 2017
Cf. A270973, A075819, A001220.

list(lim)=my(v=List(),pq); forprime(p=3,lim\6, forprime(q=3,min(lim\(2*p),p), pq=p*q; if(Mod(4,pq)^pq==2, listput(v,2*pq)))); Set(v) \\ Charles R Greathouse IV, Dec 05 2017

A333134 Positive integers k such that 11^k == 2 (mod k).

Original entry on oeis.org

1, 3, 413, 1329, 6587, 11629, 75761, 925071199, 9031140861789, 114876097917387, 1314252479257933

Offset: 1

Seiichi Manyama, Mar 20 2020

No other terms below 10^16. Some larger terms: 1584680529929001639, 15598123298097725094806152851164088027801112472240274433891889912569153113. - Max Alekseyev, Jan 07 2025

Solutions to b^n == 2 (mod n): A015919 (b=2), A276671 (b=3), A130421 (b=4), A124246 (b=5), A277401 (b=7), this sequence (b=11), A116622 (b=13), A333269 (b=17).

Cf. A015960.

for(k=1, 1e6, if(Mod(11, k)^k==2, print1(k", ")))

a(9)-a(11) from Max Alekseyev, Jan 07 2025

cp's OEIS Frontend

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

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A333269 Positive integers n such that 17^n == 2 (mod n).

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A067934 Let rep(k) = (10^k - 1)/9 be the k-th repunit number = 11111..1111 with k 1 digits, then sequence gives values of k such that rep(k) == 1 (mod k).

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A216885 Primes p such that x^47 = 2 has a solution mod p.

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A235480 Primes whose base-3 representation is also the base-9 representation of a prime.

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A165285 Primes which are the sum of at least 2 consecutive pars "Prime and PreviousNumber".

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A244149 a(n) = 2*(n*denominator(((n-1)*(n^2)+2^(n+1)-4)/(2*n))-n)/n+1.

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A294646 a(n) = (1/2)^(2*n) mod (2*n+1).

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A244149 a(n) = 2(ndenominator(((n-1)(n^2)+2^(n+1)-4)/(2n))-n)/n+1.

A294646 a(n) = (1/2)^(2n) mod (2n+1).

A296117 Base-2 pseudoprimes of the form 2pq where p and q are primes.