A006517 -id:A006517 - cp's OEIS frontend

A006521 Numbers n such that n divides 2^n + 1.

1, 3, 9, 27, 81, 171, 243, 513, 729, 1539, 2187, 3249, 4617, 6561, 9747, 13203, 13851, 19683, 29241, 39609, 41553, 59049, 61731, 87723, 97641, 118827, 124659, 177147, 185193, 250857, 263169, 292923, 354537, 356481, 373977, 531441, 555579, 752571

Offset: 1

N. J. A. Sloane

nonn

Closed under multiplication: if x and y are terms then so is x*y.
More is true: 1. If n is in the sequence then so is any multiple of n having the same prime factors as n. 2. If n and m are in the sequence then so is lcm(n,m). For a proof see the Bailey-Smyth reference. Elements of the sequence that cannot be generated from smaller elements of the sequence using either of these rules are called *primitive*. The sequence of primitive solutions of n|2^n+1 is A136473. 3. The sequence satisfies various congruences, which enable it to be generated quickly. For instance, every element of this sequence not a power of 3 is divisible either by 171 or 243 or 13203 or 2354697 or 10970073 or 22032887841. See the Bailey-Smyth reference. - Toby Bailey and Christopher J. Smyth, Jan 13 2008
A000051(a(n)) mod a(n) = 0. - Reinhard Zumkeller, Jul 17 2014
The number of terms < 10^n: 3, 5, 9, 15, 25, 40, 68, 114, 188, 309, 518, 851, .... - Robert G. Wilson v, May 03 2015
Also known as Novák numbers after Břetislav Novák who was apparently the first to study this sequence. - Charles R Greathouse IV, Nov 03 2016
Conjecture: if n divides 2^n+1, then (2^n+1)/n is squarefree. Cf. A272361. - Thomas Ordowski, Dec 13 2018
Conjecture: For k > 1, k^m == 1 - k (mod m) has an infinite number of positive solutions. - Juri-Stepan Gerasimov, Sep 29 2019

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 243, p. 68, Ellipses, Paris 2008.
R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 142.
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #16.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1064
Toby Bailey and Chris Smyth, Primitive solutions of n|2^n+1.
Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016.
V. Meally, Letter to N. J. A. Sloane, May 1975
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Subsequence of A014945.
Cf. A057719 (prime factors), A136473 (primitive n such that n divides 2^n+1).
Cf. A000051, A006517.
Cf. A066807 (the corresponding quotients).
Solutions to k^m == k-1 (mod m): 1 (k = 1), this sequence (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), A328033 (k = 7).
Column k=2 of A333429.

a006521 n = a006521_list !! (n-1)
a006521_list = filter (\x -> a000051 x `mod` x == 0) [1..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..6*10^5] | (2^n+1) mod n eq 0 ]; // Vincenzo Librandi, Dec 14 2018

for n from 1 to 1000 do if 2^n +1 mod n = 0 then lprint(n); fi; od;
S:=1,3,9,27,81:C:={171,243,13203,2354697,10970073,22032887841}: for c in C do for j from c to 10^8 by 2*c do if 2&^j+1 mod j = 0 then S:=S, j;fi;od;od; S:=op(sort([op({S})])); # Toby Bailey and Christopher J. Smyth, Jan 13 2008

Do[If[PowerMod[2, n, n] + 1 == n, Print[n]], {n, 1, 10^6}]
k = 9; lst = {1, 3}; While[k < 1000000, a = PowerMod[2, k, k]; If[a + 1 == k, AppendTo[lst, k]]; k += 18]; lst (* Robert G. Wilson v, Jul 06 2009 *)
Select[Range[10^5], Divisible[2^# + 1, #] &] (* Robert Price, Oct 11 2018 *)

for(n=1,10^6,if(Mod(2,n)^n==-1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014

A006521_list = [n for n in range(1,10**6) if pow(2,n,n) == n-1] # Chai Wah Wu, Jul 25 2017

More terms from David W. Wilson, Jul 06 2009

A015891 Numbers k such that k | 5^k + 5.

Original entry on oeis.org

1, 2, 5, 6, 10, 30, 70, 1565, 2806, 3126, 51670, 58290, 214405, 285286, 378258, 1854766, 2170486, 2222122, 2247610, 3463230, 4147522, 5942526, 9381126, 14818486, 15743890, 20162858, 34087054, 34838686, 38742166, 71067430

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A067946 = numbers n such that n divides 5^n-1. Cf. A015951 = numbers n such that n | 5^n + 1.

Cf. A006517, A015888, A015889, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Select[Range[1000000], IntegerQ[(PowerMod[5,#,# ]+5)/# ]&] (* Alexander Adamchuk *)

Corrected by Alexander Adamchuk, Nov 04 2006

A055685 Numbers k such that 2^k == -1 (mod k-1).

Original entry on oeis.org

2, 6, 66, 378, 1190, 1470, 25806, 58590, 134946, 137346, 170586, 272610, 285390, 420210, 538734, 592410, 618450, 680706, 778806, 1163066, 1520442, 1700946, 2099202, 2831010, 4020030, 4174170, 4516110, 5059890, 5215770

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

The sequence is infinite. In fact, even its intersection with A006517 (given by A219037) is infinite.

Chai Wah Wu, Table of n, a(n) for n = 1..1192
Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.

Cf. A064935, A006517, A219037, A296369.

Do[If[PowerMod[2, n, n-1]==n-2, Print[n]], {n, 2, 12900000}]

A055685_list = [n for n in range(2,10**6) if pow(2,n,n-1) == n-2] # Chai Wah Wu, Nov 04 2019

a(n) = A296369(n) + 1.

Edited by Max Alekseyev, Oct 11 2012

Incorrect term 4285390 removed by Chai Wah Wu, Nov 04 2019

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

Original entry on oeis.org

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

A128122 Numbers m such that 2^m == 6 (mod m).

Original entry on oeis.org

1, 2, 10669, 6611474, 43070220513807782

Offset: 1

Alexander Adamchuk, Feb 15 2007

No other terms below 10^17. - Max Alekseyev, Nov 18 2022

A large term: 862*(2^861-3)/281437921287063162726198552345362315020202285185118249390789 (203 digits). - Max Alekseyev, Sep 24 2016

			2 == 6 (mod 1), so 1 is a term;
4 == 6 (mod 2), so 2 is a term.

Joe K. Crump, 2^n mod n
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A000079 (k=0),A187787 (k=1/2), A296369 (k=-1/2), A006521 (k=-1), A296370 (k=3/2), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), this sequence (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)

Cf. A015910, A036236.

m = 6; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

1 and 2 added by N. J. A. Sloane, Apr 23 2007

a(5) from Max Alekseyev, Nov 18 2022

A015888 Numbers k such that k | (3^k + 3).

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 15, 28, 366, 435, 532, 804, 1036, 4636, 6364, 7932, 11476, 15051, 19684, 40132, 75396, 80956, 247395, 266476, 280756, 307396, 356644, 490372, 531444, 544348, 557956, 565516, 572716, 808228, 1007574, 1256356, 1695484

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..400

Cf. A006517, A015889, A015891, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Join[{1,2},Select[Range[17*10^5],PowerMod[3,#,#]==#-3&]] (* Harvey P. Dale, Mar 31 2024 *)

Terms 1, 2 inserted by Max Alekseyev, Nov 24 2017

A015889 Numbers k that divide 4^k + 4.

Original entry on oeis.org

1, 2, 4, 10, 20, 52, 130, 370, 1066, 1090, 2626, 4580, 7540, 9140, 21268, 21682, 24634, 26596, 29380, 38740, 40330, 65026, 146810, 152164, 154180, 159172, 211492, 271426, 290116, 330706, 432730, 567988, 575956, 710788, 782090

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..750

Cf. A006517, A015888, A015891, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Select[Range[800000],Divisible[4^#+4,#]&] (* Harvey P. Dale, Jun 06 2012 *)

Corrected (a(2) = 2 inserted) by Harvey P. Dale, Jun 06 2012

A015892 Numbers k such that k | 6^k + 6.

Original entry on oeis.org

1, 2, 3, 6, 14, 42, 66, 111, 606, 742, 3486, 6666, 8646, 13118, 26751, 58618, 66886, 125686, 141526, 144886, 489958, 525994, 695254, 752691, 1198554, 1234806, 2200386, 2479686, 2671306, 2862006, 3277626, 3716602, 5923126, 6490947

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..200

Cf. A006517, A015888, A015889, A015891, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Select[Range[10^5], Mod[PowerMod[6, #, #] + 6, #] == 0 &] (* Giovanni Resta, Oct 23 2018 *)

Missing a(2)-a(3) from Giovanni Resta, Oct 23 2018

A015893 Numbers k such that k | 7^k + 7.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 28, 35, 56, 91, 172, 175, 616, 742, 904, 946, 1204, 1267, 1288, 2275, 2486, 4795, 5551, 7288, 10396, 10696, 16426, 16471, 18536, 31675, 34126, 38872, 40132, 40775, 58618, 61075, 67768, 72136, 78604, 86296, 117656

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1200

Cf. A006517, A015888, A015889, A015891, A015892, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Select[Range[10^5], Mod[PowerMod[7, #, #] + 7, #] == 0 &] (* Giovanni Resta, Oct 23 2018 *)

Missing a(2)-a(3) from Giovanni Resta, Oct 23 2018

A015897 Numbers k such that k | 8^k + 8.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 24, 36, 66, 72, 76, 344, 396, 804, 946, 1548, 1986, 3336, 5526, 7596, 8196, 8646, 10116, 11916, 21846, 25156, 34056, 43352, 47972, 51426, 66788, 117522, 131076, 180246, 195084, 199606, 250284, 265826, 307876, 319404

Offset: 1

Robert G. Wilson v

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..733

Cf. A006517, A015888, A015889, A015891, A015892, A015893, A015898, A015902, A015903, A015904, A015905, A015906.

is(n)=Mod(8,n)^n==-8 \\ Charles R Greathouse IV, Oct 08 2015

a(2)-a(4) inserted by Charles R Greathouse IV, Oct 08 2015

cp's OEIS Frontend

A006521 Numbers n such that n divides 2^n + 1.

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A015891 Numbers k such that k | 5^k + 5.

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A055685 Numbers k such that 2^k == -1 (mod k-1).

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

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A128122 Numbers m such that 2^m == 6 (mod m).

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A015888 Numbers k such that k | (3^k + 3).

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A015889 Numbers k that divide 4^k + 4.

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A015892 Numbers k such that k | 6^k + 6.

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