A057719 -id:A057719 - 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

A171980 Prime divisors of elements of A129066.

Original entry on oeis.org

5, 3001, 120041, 532501, 720241, 2160721, 3937501, 9375001, 16505501, 120040001, 158453021, 165055001, 202567501, 289312501, 562500061, 900307501, 985937501, 1500512501, 1512504701, 3169060421, 3301100021, 3908604433, 3993757501

Offset: 1

Max Alekseyev, Jan 20 2010

nonn

Corresponding smallest multiples from A129066 are given in A171981.

Prime p>5 is in this sequence if the multiplicative order of (sqrt(5)-3)/2 modulo p is the product of smaller terms of this sequence.

Max Alekseyev, Table of n, a(n) for n = 1..25 (complete up to 10^10)

Cf. A057719, A066364, A129729, A171981

A087807 Prime factors of solutions to 24^n == 1 (mod n).

Original entry on oeis.org

23, 47, 14759, 49727, 124799, 304751, 497261, 609503, 1828507, 2685259, 10741037, 12872687, 13877879, 23462213, 23652649, 27755759, 29134267, 31908959, 53753807, 65205263, 132771091, 218148653, 341965703, 551361983, 734951759

Offset: 1

Thomas Baruchel, Oct 14 2003

Primes that divide at least one term of A014960.

Prime p is in this sequence iff the multiplicative order of 24 modulo p is the product of smaller terms of this sequence. - Max Alekseyev, May 26 2010

			A014960(12) = 2870377 = 23 * 124799

Cf. A014960, A057719, A171980, A134360, A136372, A136373, A129729, A066364.

Corrected and extended by Max Alekseyev, May 26 2010

Edited by Max Alekseyev, Nov 16 2019

A136474 Primes that divide 2^(3^n)+1 for some n.

Original entry on oeis.org

3, 19, 163, 1459, 17497, 52489, 87211, 135433, 139483, 1220347, 5419387, 6049243, 28934011, 86093443, 227862073, 272010961

Offset: 1

Christopher J. Smyth, Feb 16 2008

This sequence is a subsequence of A057719.

272010961 is the last term less than 3*10^9. The n for each prime is 0, 2, 4, 5, 7, 8, 3, 4, 5, 9, 7, 7, 8, 16, 6, 4. Some terms from A111974 are in this sequence also: 411782264189299, 116299474006080119380780339, and 84782316550432407028588866403. If p=2*3^k+1 is prime for an even k, then p is in this sequence.

			1220347 belongs to the sequence as it is a factor of 2^(3^9)+1 (This is the largest member of the sequence less than 5000000)

Cf. A057719, A136475, A005621, A136473.

with(numtheory):L:=3;for p from 5 to 5000000 do if isprime(p) then q:=op(2,ifactors(order(2,p)));if nops(q)=2 then if op(1,op(1,q))=2 and op(2,op(1,q))=1 and op(1,op(2,q))=3 then L:=L,p;fi;fi;fi;od;L;

Reap[Do[p=Prime[n]; mo=MultiplicativeOrder[2, p]; If[EvenQ[mo] && IntegerQ[Log[3,mo/2]], Sow[p]], {n, PrimePi[10^7]}]][[2,1]]

A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

Original entry on oeis.org

3, 17, 103, 163, 373, 487, 1733, 3469, 4373, 8803, 10259, 15607, 16069, 26237, 26297, 31193, 31517, 35153, 37987, 38047, 38149, 39367, 52817, 60427, 60589, 61553, 74357, 76837, 78713, 100733, 103979, 114377, 119891, 152189, 181277, 231131, 235891, 238307, 239783, 280927, 289243, 316903, 338581

Offset: 1

Jonathan Vos Post, Aug 26 2009

nonn

This is the last sequence on p. 15 of Smyth. [WARNING: Smyth lists 2 as a possible prime factor, which, in fact, is not possible. - Max Alekseyev, Sep 17 2024]
The Lucas sequence with P = 3, Q = 5 is defined as v=2,3,-1,-18,-49,-57,.. where v(n) = P*v(n-1)-Q*v(n-2), with g.f. (2-3x)/(1-3x+5x^2).
The indices n such that n|v(n) define the sequence T = 1,3,9,27,81,153,243,459,... as listed by Smyth.
The OEIS sequence shows all distinct prime factors of elements of T.

Max Alekseyev, Table of n, a(n) for n = 1..627
Richard André-Jeannin, Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart., 29(4) (1991) 364-366.
Yu. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001) 75-122.
R. D. Carmichael, On the numerical factors of the arithmetic forms alpha*n+-beta*n, Annals of Math., 2nd ser., 15 (1/4) (1913/14) 30-48.
Chris Smyth, The Terms in Lucas Sequences Divisible by their Indices, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4. Preprint: arXiv:0908.3832 [math.NT], 2009.

Cf. A057719, A066364, A087807.

More detailed definition, comments rephrased, non-ascii characters in URL's removed - R. J. Mathar, Sep 09 2009
a(8)-a(9), a(11), a(18) from Jean-François Alcover, Dec 08 2017
Incorrect codes (depending on a search limit) removed, prime 2 removed, terms a(10), (12)-a(17), and a(19) onward added by Max Alekseyev, Sep 17 2024

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

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A171980 Prime divisors of elements of A129066.

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A087807 Prime factors of solutions to 24^n == 1 (mod n).

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A136474 Primes that divide 2^(3^n)+1 for some n.

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A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

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A289258 Prime factors of numbers in A289257.

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