A055685 -id:A055685 - cp's OEIS frontend

A006517 Numbers k such that k divides 2^k + 2.

1, 2, 6, 66, 946, 8646, 180246, 199606, 265826, 383846, 1234806, 3757426, 9880278, 14304466, 23612226, 27052806, 43091686, 63265474, 66154726, 69410706, 81517766, 106047766, 129773526, 130520566, 149497986, 184416166, 279383126

Offset: 1

N. J. A. Sloane, David W. Wilson

All terms greater than 1 are even. If an odd term n>1 exists then n = m*2^k + 1 for some k>=1 and odd m. Then n divides 2^(m*2^k) + 1 and so does every prime factor p of n, implying that 2^(k+1) divides the multiplicative order of 2 modulo p and thus p-1. Therefore n = m*2^k + 1 is the product of prime factors of the form t*2^(k+1) + 1, implying that n-1 is divisible by 2^(k+1), a contradiction. - Max Alekseyev, Mar 16 2009
The sequence is infinite. In fact, its intersection with A055685 (given by A219037) is infinite (see Li et al. link). - Max Alekseyev, Oct 11 2012
All terms greater than 6 have at least three distinct prime factors. - Robert Israel, Aug 21 2014

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 142.
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #18
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..150
Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.
V. Meally, Letter to N. J. A. Sloane, May 1975

Cf. A006521, A015888, A015889, A015891, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Do[ If[ PowerMod[ 2, n, n ] + 2 == n, Print[n]], {n, 2, 1500000000, 4} ]
Join[{1},Select[Range[28*10^7],PowerMod[2,#,#]==#-2&]] (* Harvey P. Dale, Aug 13 2018 *)

is_A006517(n)=!(Mod(2,n)^n+2)  \\ M. F. Hasler, Oct 08 2012

Corrected and extended by Joe K. Crump (joecr(AT)carolina.rr.com), Sep 12 2000 and Robert G. Wilson v, Sep 13 2000

A187787 Numbers k such that 2^(k+1) == 1 (mod k).

Original entry on oeis.org

1, 3, 15, 35, 119, 255, 455, 1295, 2555, 2703, 3815, 3855, 4355, 5543, 6479, 8007, 9215, 10439, 10619, 11951, 16211, 22895, 23435, 26319, 26795, 27839, 28679, 35207, 43055, 44099, 47519, 47879, 49679, 51119, 57239, 61919, 62567, 63167, 63935, 65535, 74447, 79055

Offset: 1

Franz Vrabec, Jan 06 2013

nonn

Prime factorizations of the first ten terms: 3, 3*5, 5*7, 7*17, 3*5*17, 5*7*13, 5*7*37, 5*7*73, 3*17*53, 5*7*109.

			3 is in the sequence because 2^(3+1) mod 3 = 16 mod 3 = 1.

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

Cf. A055685, A069927, A112983.

for n from 1 to 100000 do if 2&^(n+1) mod n = 1 then print(n) fi od;

m = 1; Join[Select[Range[1, m], Divisible[2^(# + 1), #] &],
Select[Range[m + 1, 10^5], PowerMod[2, # + 1, #] == m &]] (* Robert Price, Oct 11 2018 *)
Join[{1},Select[Range[80000],PowerMod[2,#+1,#]==1&]] (* Harvey P. Dale, Aug 19 2019 *)

for (n=1,10^7, if (Mod(2,n)^(n+1)==1,print1(n,", "))); /* Joerg Arndt, Jan 06 2013 */

Term a(1)=1 prepended by Max Alekseyev, Nov 29 2014

A296369 Numbers m such that 2^m == -1/2 (mod m).

Original entry on oeis.org

1, 5, 65, 377, 1189, 1469, 25805, 58589, 134945, 137345, 170585, 272609, 285389, 420209, 538733, 592409, 618449, 680705, 778805, 1163065, 1520441, 1700945, 2099201, 2831009, 4020029, 4174169, 4516109, 5059889, 5215769

Offset: 1

Max Alekseyev, Dec 10 2017

nonn

Equivalently, 2^(m+1) == -1 (mod m), or m divides 2^(m+1) + 1.

The sequence is infinite, see A055685.

Chai Wah Wu, Table of n, a(n) for n = 1..1192
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A296370 (k=3/2), A187787 (k=1/2), this sequence (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[10^5], Divisible[2^(# + 1) + 1, #] &] (* Robert Price, Oct 11 2018 *)

A296369_list = [n for n in range(1,10**6) if pow(2,n+1,n) == n-1] # Chai Wah Wu, Nov 04 2019

a(n) = A055685(n) - 1.

Incorrect term 4285389 removed by Chai Wah Wu, Nov 04 2019

A055686 Numbers k such that 3^k == -1 (mod k-1).

Original entry on oeis.org

2, 3, 5, 6, 15, 245, 366, 435, 855, 2295, 3795, 5967, 7875, 10878, 26475, 33915, 117615, 188775, 231435, 284355, 487635, 501039, 589155, 593775, 621675, 755595, 1255815, 1306935, 1642095, 1911195, 2193125, 2434755, 2484675, 2507835

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A055685, A055687, A055688, A055689, A055690, A055691, A055692, A055693, A055694, A055695.

Do[If[PowerMod[3, n, n-1]==n-2, Print[n]], {n, 2, 10^7}]

A328230 Numbers m that divide 3^(m + 1) + 1.

Original entry on oeis.org

1, 2, 4, 5, 14, 244, 365, 434, 854, 2294, 3794, 5966, 7874, 10877, 26474, 33914, 117614, 188774, 231434, 284354, 487634, 501038, 589154, 593774, 621674, 755594, 1255814, 1306934, 1642094, 1911194, 2193124, 2434754, 2484674, 2507834, 2621654, 2643494, 3512114, 3759854, 3997574, 4082246

Offset: 1

Juri-Stepan Gerasimov, Oct 08 2019

nonn

Conjecture: For k > 2, k^(m + 1) == -1 (mod m) has an infinite number of positive solutions.

Chai Wah Wu, Table of n, a(n) for n = 1..1528 (n = 1..100 from Robert Israel)

Cf. A055685, A277289, A296369.

[n+1: n in [0..5000000] | Modexp(3,n+2,n+1) eq n];

filter:= m -> 3 &^ (m+1) + 1 mod m = 0:
select(filter, [$1..10^7]); # Robert Israel, Oct 30 2019

isok(m) = Mod(3, m)^(m+1) == -1; \\ Michel Marcus, Oct 10 2019

A055688 Numbers k such that 5^k == -1 (mod k-1).

Original entry on oeis.org

2, 3, 7, 14, 15, 175, 855, 2695, 78127, 103974, 106695, 121975, 420210, 487375, 1299375, 2174655, 3895095, 4151455, 5842215, 5951130, 6508335, 10637055, 20117895, 24482958, 31999695, 32282054, 32620203, 32872455, 34258455

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..200

Cf. A055685, A055686, A055687, A055689, A055690, A055691, A055692, A055693, A055694, A055695.

Do[If[PowerMod[5, n, n-1]==n-2, Print[n]], {n, 2, 10^8}]
Select[Range[2,35*10^6],PowerMod[5,#,#-1]==#-2&] (* Harvey P. Dale, Aug 13 2024 *)

a(22) corrected by Amiram Eldar, Jul 23 2021

A055690 Numbers k such that 7^k == -1 (mod k-1).

Original entry on oeis.org

2, 3, 5, 6, 9, 26, 45, 66, 87, 345, 765, 906, 3926, 8405, 11766, 23805, 35145, 42966, 59685, 95289, 317250, 413325, 416757, 722745, 770066, 890825, 938457, 1325826, 1921986, 3315378, 3675555, 5299250, 6791445, 6899685, 9371826, 10892313

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..285

Cf. A055685, A055686, A055687, A055688, A055689, A055691, A055692, A055693, A055694, A055695.

Do[If[PowerMod[7, n, n-1]==n-2, Print[n]], {n, 2, 2*10^7}]

A064935 Numbers k such that (k+3)^(k+2) mod (k+1) = k.

Original entry on oeis.org

4, 64, 376, 1188, 1468, 25804, 58588, 134944, 137344, 170584, 272608, 285388, 420208, 538732, 592408, 618448, 680704, 778804, 1163064, 1520440, 1700944, 2099200, 2831008, 4020028, 4174168, 4516108, 5059888, 5215768, 5447272

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 26 2001

nonn

From Robert Israel, Feb 13 2025: (Start)
  Numbers k such that 2^(k+2) == -1 (mod k+1).
  All terms are divisible by 4.
  The only term k where k+1 is prime is 4.
(End)

			(4+3)^(4+2) mod (4+1) = 7^6 mod 5 = 117649 mod 5 = 4, so 4 is a term.

Robert Israel, Table of n, a(n) for n = 1..350

Equals A055685(n+1) - 2.

filter:= proc(k) 2 &^(k+2) mod (k+1) = k end proc:
select(filter, [seq(i,i=4..10^7,4)]); # Robert Israel, Feb 13 2025

isok(k) = Mod(k+3, k+1)^(k+2) == k; \\ Michel Marcus, Jul 12 2021

A219037 Numbers k such that k divides 2^k + 2 and (k-1) divides 2^k + 1.

Original entry on oeis.org

2, 6, 66, 73786976294838206466

Offset: 1

Max Alekseyev, Nov 10 2012

Also, numbers k such that 2^k == k-2 (mod k*(k-1)).
The sequence is infinite: if m is in this sequence, then so is 2^m + 2.
No other terms below 10^20.

W. Sierpinski, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #18.

Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.

Intersection of A006517 and A055685.

Cf. A217468, A216822, A171959.

Conjecture: a(n+1) = 2^a(n) + 2 for all n.

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.

cp's OEIS Frontend

A006517 Numbers k such that k divides 2^k + 2.

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A187787 Numbers k such that 2^(k+1) == 1 (mod k).

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A296369 Numbers m such that 2^m == -1/2 (mod m).

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A055686 Numbers k such that 3^k == -1 (mod k-1).

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A328230 Numbers m that divide 3^(m + 1) + 1.

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Magma

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A055688 Numbers k such that 5^k == -1 (mod k-1).

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A055690 Numbers k such that 7^k == -1 (mod k-1).

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A064935 Numbers k such that (k+3)^(k+2) mod (k+1) = k.

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