A069051 -id:A069051 - cp's OEIS frontend

A014741 Numbers k such that k divides 2^(k+1) - 2.

1, 2, 6, 18, 42, 54, 126, 162, 294, 342, 378, 486, 882, 1026, 1134, 1314, 1458, 1806, 2058, 2394, 2646, 3078, 3402, 3942, 4374, 5334, 5418, 6174, 6498, 7182, 7938, 9198, 9234, 10206, 11826, 12642, 13122, 14154, 14406, 16002, 16254

Offset: 1

N. J. A. Sloane, Olivier Gérard

nonn

Also, numbers k such that k divides Eulerian number A000295(k+1) = 2^(k+1) - k - 2.
Also, numbers k such that k divides A086787(k) = Sum_{i=1..k} Sum_{j=1..k} i^j.
All terms greater than 1 are even; for a proof, see comment in A036236. - Max Alekseyev, Feb 03 2012
If k>1 is a term, then 3*k is also a term. - Alexander Adamchuk, Nov 03 2006
Prime numbers of the form a(m)+1 are given by A069051. - Max Alekseyev, Nov 14 2012
The number 2^m - 2 is a term of this sequence if and only if m - 1 is a term. - Thomas Ordowski, Jul 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000295, A086787, A015919, A006517, A330382.

Join[{1,2},Select[Range[17000],PowerMod[2,#+1,#]==2&]] (* Harvey P. Dale, Feb 11 2015 *)

is(n)=Mod(2,n)^(n+1)==2 \\ Charles R Greathouse IV, Nov 03 2016

For n > 1, a(n) = 2*A014945(n-1). - Max Alekseyev, Nov 14 2012

A216822 Numbers n such that 2^n == 2 (mod n*(n+1)).

Original entry on oeis.org

1, 5, 13, 29, 37, 61, 73, 157, 181, 193, 277, 313, 397, 421, 457, 541, 561, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1289, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1905, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917

Offset: 1

V. Raman, Sep 17 2012

a(17) = 561 is the first composite number in the sequence. - Charles R Greathouse IV, Sep 19 2012

Intersection of { A015919(n) } and { A192109(n)-1 }. - Max Alekseyev, Apr 22 2013

V. Raman and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A069051 (prime n such that 2^n == 2 (mod n*(n-1))).
Cf. A217466 (prime terms of the sequence).
Cf. A217465 (composite terms of the sequence)

Select[Range[1, 10000], Mod[2^# - 2, # (# + 1)] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{1},Select[Range[3000],PowerMod[2,#,#(#+1)]==2&]] (* Harvey P. Dale, Oct 05 2022 *)

is(n)=Mod(2,n*(n+1))^n==2; \\ Charles R Greathouse IV, Sep 19 2012

A216822_list = [n for n in range(1,10**6) if n == 1 or pow(2,n,n*(n+1)) == 2] # Chai Wah Wu, Mar 25 2021

a(1)=1 prepended by Max Alekseyev, Dec 29 2017

A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

Original entry on oeis.org

91625794219, 8796093022207, 1557609722332488343, 18216643597893471403

Offset: 1

V. Raman, Oct 04 2012

No other terms below 2^64.
The next term is <= 25790417485109157029391019 = A019320(258). - Emmanuel Vantieghem, Dec 01 2014
Terms A019320(k) belongs to this sequence for k in A297412. - Max Alekseyev, Dec 29 2017

			a(1) = 91625794219 = (2^38 - 2^19 + 1)/3 = A019320(114).
a(2) = 8796093022207 = 2^43 - 1 = A019320(43).

Mersenne Forum, Prime Conjecture

Cf. A069051, A216822, A219037, A217465.

Intersection of A001567 and { 2*A014945(k) + 1 }.

is_A217468(n)={Mod(2,(n-1)*n)^n==2 & !isprime(n)}  \\ - M. F. Hasler, Oct 07 2012

A211349 Primes p such that p-1 divides 2^p + 2.

Original entry on oeis.org

2, 3, 11, 251, 5051, 16811, 2025251, 8751251, 16607051, 28257611, 69005051, 78906251, 176775251, 210381251, 372175451, 550427051, 707025251, 854704451, 1866788051, 2441406251, 2605806251, 4249701251, 5469531251, 9304386251, 10315761251, 10915095251

Offset: 1

Philip A. Hoskins, Feb 06 2013

nonn

Prime p>2 is in this sequence iff (p-1)/2 is in A015950. - Max Alekseyev, Dec 26 2017

Max Alekseyev, Table of n, a(n) for n = 1..177

Cf. A069051, A211203.

Select[Prime[Range[1000]], Mod[2^# + 2, # - 1] == 0 &]

N=10^9;
default(primelimit,N);
forprime(p=2,N, if (-2==Mod(2,p-1)^p, print1(p,", ")));
/* Joerg Arndt, Feb 06 2013 */

from sympy import primerange
A211349_list = [p for p in primerange(1,10**6) if p == 2 or pow(2,p,p-1) == p-3] # Chai Wah Wu, Mar 25 2021

a(19)-a(47) from Giovanni Resta, Feb 10 2013

a(48)-a(177) from Max Alekseyev, Jan 06 2018

A211203 Prime numbers p such that p-1 divides (2^(p-1)+1)*(2^p-2).

Original entry on oeis.org

2, 3, 7, 11, 19, 31, 43, 79, 127, 151, 163, 211, 251, 271, 311, 331, 379, 487, 547, 631, 751, 811, 883, 991, 1051, 1171, 1231, 1459, 1471, 1831, 1951, 1999, 2251, 2311, 2531, 2647, 2731, 2791, 2971, 3079, 3331, 3511, 3631, 3691, 3823, 3943, 4051, 4447, 4651

Offset: 1

Philip A. Hoskins, Feb 06 2013

nonn

This is also the set of primes such that n^(4^(p-1)) is congruent to n or -n modulo p.

Prime p>2 is in this sequence iff (p-1)/2 belongs to A014957. - Max Alekseyev, Dec 26 2017

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..204 from Philip A. Hoskins)

Cf. A069051 (primes p such that p - 1 divides 2^p - 2)

Cf. A211349 (primes p such that p - 1 divides 2^p + 2)

A211203:=proc(q)
local n;
for n from 1 to q do
  if type((2^(2*ithprime(n)-1)-2)/(ithprime(n)-1),integer) then print(ithprime(n));
fi; od; end:
A211203(10000000); # Paolo P. Lava, Feb 18 2013

Select[Prime[Range[1000]], Mod[1/2*(2^# + 2)*(2^# - 2), # - 1] == 0 &]

is(p) = lift((Mod(2,p-1)^(p-1)+1)*(Mod(2,p-1)^p-2))==0 \\ David A. Corneth, Mar 25 2021

from sympy import primerange
A211203_list = [p for p in primerange(1,10**6) if p == 2 or p == 3 or pow(2,2*p-1,p-1) == 2] # Chai Wah Wu, Mar 25 2021

A330382 Composite numbers k such that k-1 divides 2^k-2.

Original entry on oeis.org

55, 295, 343, 1027, 1135, 1315, 1807, 2059, 2395, 3403, 4375, 5335, 6175, 6499, 7183, 7939, 9235, 10207, 12643, 13123, 14155, 16003, 16255, 19495, 21547, 23815, 27595, 27703, 30619, 35479, 37927, 43219, 45487, 48007, 48763, 50275, 55567, 58483, 64387, 64639, 74899

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 12 2019

nonn

If k is in this sequence, then 2^k-1 is also a term, so this sequence is infinite.
Also 2^p-1 is in this sequence for such prime p in A069051 that 2^p-1 is composite.
Theorem: if k-1 | 2^k-2, then m-1 | 2^m-2, where m = 2^k-1.
Conjecture: k-1 | 2^k-2 for k = (2^n-1)^3 if and only if n(n-1) | 2^n-2 for n > 2.
It seems that A007013(n)^3 for n > 1 and A007013(n) for n > 4 are in this sequence.
These are the composites k for which M - 1 divides 2^M - 2 where M = 2^k - 1. - Thomas Ordowski, Jul 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A007013, A014741, A054723, A069051, A217468.

A217468 is a subsequence.

Select[Range[75000], CompositeQ[#] && Divisible[PowerMod[2, #, # - 1] - 2, # - 1] &]

forcomposite(k=1,75000,if(!((2^k-2)%(k-1)),print1(k,", "))) \\ Hugo Pfoertner, Dec 12 2019

Composites of A014741(n) + 1. - Thomas Ordowski, Jul 01 2024

A350176 Numbers m such that binomial(m, 3) divides binomial(3^m-2, 3).

Original entry on oeis.org

3, 5, 7, 17, 79, 97, 257, 457, 65537, 677041, 1354081, 7812169, 13650001, 21381361, 65246161, 134246401, 242235841, 277032001, 393414001, 468930001, 793605121, 859560241, 886966081, 1609179001, 3355067041, 4269249601, 6024794161, 8120048641, 10142988241, 10466887201

Offset: 1

Juri-Stepan Gerasimov, Dec 18 2021

nonn

Are all terms prime numbers?

Chai Wah Wu, Table of n, a(n) for n = 1..40

Supersequence of A019434.

Cf. A069051 (binomial(k,2) divides binomial(2^k-1, 2)?).

[n: n in [3..10^4] |  IsZero(Binomial(3^n-2, 3) mod Binomial(n,3))];

Select[Range[3, 10^5], Divisible[Binomial[3^# - 2, 3], Binomial[#, 3]] &] (* Amiram Eldar, Dec 18 2021 *)

isok(m) = if (m>=3, (binomial(3^m-2, 3) % binomial(m, 3)) == 0); \\ Michel Marcus, Dec 19 2021

isok(m) = if (m>2, my(md = Mod(3, m^3 - 3*m^2 + 2*m)^m); (md^3 - 9*md^2 + 26*md - 24) == 0); \\ Michel Marcus, Dec 28 2021

from itertools import count, islice
def A350176_gen(startvalue=3): # generator of terms >= startvalue
    for m in count(max(startvalue,3)):
        k = m*(m-1)*(m-2)
        a = pow(3,m,k)-2
        if (a*(a-1)*(a-2))%k == 0:
            yield m
A350176_list = list(islice(A350176_gen(),10)) # Chai Wah Wu, Jul 21 2025

a(11)-a(20) from Michel Marcus, Dec 27 2021
a(21)-a(23) from Michel Marcus, Dec 28 2021
a(24)-a(30) from Chai Wah Wu, Jul 22 2025

A374841 Numbers k such that 2^(2^k-2) == 1 (mod k^2).

Original entry on oeis.org

3, 7, 19, 43, 73, 127, 163, 337, 341, 379, 487, 601, 881, 883, 937, 1387, 1459, 1801, 2593, 2647, 2857, 3079, 3529, 3673, 3943, 4057, 4201, 4681, 5419, 5461, 5881, 6121, 6481, 6529, 6553, 6571, 6841, 7481, 7993, 8233, 8911, 9001, 9199, 9241, 9721, 10261, 10657, 11161, 11827, 12241

Offset: 1

Thomas Ordowski, Jul 22 2024

nonn

If p is an odd prime and 2^(2^p-2) == 1 (mod p), then 2^(2^p-2) == 1 (mod p^2).
If 2^(k-1) == 1 (mod k) and 2^(2^k-2) == 1 (mod k), then 2^(2^k-2) == 1 (mod k^2).
Composite terms that are not Fermat pseudoprimes to base 2 are 66709, 951481, ...
Note that 66709 = 19*3511 and 951481 = 271*3511, where 3511 is a Wieferich prime.

			3 is a term, because 3^2 divides 2^(2^3-2) - 1 = 2^6 - 1 = 63.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..250 from Michel Marcus)

Cf. A001220, A001567, A069051 (> 2 is a subsequence), A217468 (subsequence).

Select[Range[12500], PowerMod[2, 2^# - 2, #^2] == 1 &] (* Amiram Eldar, Jul 22 2024 *)

isok(k) = Mod(2, k^2)^(2^k-2) == 1; \\ Michel Marcus, Jan 05 2025

More terms from Amiram Eldar Jul 22 2024

A297413 Numbers k such that 2^m == 2 (mod m*(m-1)), where m=A019320(k).

Original entry on oeis.org

2, 3, 6, 7, 14, 19, 38, 42, 43, 86, 114, 127, 163, 254, 258, 326, 379, 487, 758, 762, 883, 974, 978, 1459, 1766, 2274, 2647, 2918, 2922, 3079, 3943, 5294, 5298, 5419, 6158, 7886, 8754, 9199, 10838, 11827, 14407, 15882, 16759, 18398, 18474, 18523, 23654, 23658, 24967, 26407, 28814, 32514, 33518, 37046, 37339, 39367

Offset: 1

Max Alekseyev, Dec 29 2017

nonn

Also, numbers k such that A019320(k) belongs to A069051 or A217468.

Cf. A069051, A217468, A297414.

Contains A297412 as a subsequence.

is_A297413(k) = my(m=polcyclo(k,2)); Mod(2,m*(m-1))^m==2;

A350402 Numbers k such that binomial(k, 2) divides binomial(2^k-2, 2).

Original entry on oeis.org

2, 3, 7, 11, 19, 31, 43, 127, 163, 211, 271, 311, 331, 379, 487, 571, 631, 811, 883, 991, 1459, 1471, 1747, 2311, 2531, 2647, 2791, 2971, 3079, 3631, 3943, 4091, 5171, 5419, 6571, 7591, 8863, 8911, 9199, 9791, 9931, 10891, 11827, 11971, 13591, 14407, 15391, 16759, 17011, 18523, 19531, 21871, 22111, 23431, 24967

Offset: 1

Juri-Stepan Gerasimov, Dec 29 2021

nonn

Conjecture: aside from the first term, this is a subsequence of A094179 (numbers congruent to 3 mod 4 which are divisible only by primes congruent to 3 mod 4).

The conjecture is false: a(2295) = 508606771 = 19531 * 26041 is not in A094179, nor even A004614. - Charles R Greathouse IV, Jan 22 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Supersequence of A069051.

Cf. A069051 (binomial(k,2) divides binomial(2^k-1, 2)?), A094179, A350176.

[n: n in [2..25000] |  IsZero(Binomial(2^n-2, 2) mod Binomial(n, 2))];

Select[Range[2, 25000], Divisible[Binomial[2^# - 2, 2], Binomial[#,2]] &] (* Amiram Eldar, Dec 29 2021 *)

isok(n) = (n>1) && ((binomial(2^n-2, 2) % binomial(n, 2)) == 0); \\ Michel Marcus, Jan 04 2022

is(n)=my(m=n^2-n,t=Mod(2,m)^n-2); t*(t-1)==0 \\ Charles R Greathouse IV, Jan 20 2022

cp's OEIS Frontend

A014741 Numbers k such that k divides 2^(k+1) - 2.

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A216822 Numbers n such that 2^n == 2 (mod n*(n+1)).

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A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

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A211349 Primes p such that p-1 divides 2^p + 2.

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A211203 Prime numbers p such that p-1 divides (2^(p-1)+1)*(2^p-2).

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A330382 Composite numbers k such that k-1 divides 2^k-2.

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A350176 Numbers m such that binomial(m, 3) divides binomial(3^m-2, 3).

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