A006521 -id:A006521 - cp's OEIS frontend

A328033 Numbers m that divide 7^m + 6.

1, 13, 793, 1943, 150341, 183793, 2348789, 26052527, 27982637, 54789869, 1588344433, 3928538029, 8115802931, 16936276919, 17786709541, 47778790033, 973094452518029

Offset: 1

Juri-Stepan Gerasimov, Oct 02 2019

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

Also includes 2073273696480171732497. - Giovanni Resta, Oct 04 2019

Solutions to k^m == 1-k (mod m): A006521 (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), this sequence (k = 7), A327468 (k = 8).

Cf. A253210 (7^n + 6).

[1] cat [n: n in [1..10^8] | Modexp(7, n, n) + 6 eq n];

a(12)-a(16) from Giovanni Resta, Oct 04 2019

a(17) from Max Alekseyev, Feb 07 2024

A357125 Positive integers n such that 2^(n-3) == -1 (mod n).

Original entry on oeis.org

1, 5, 4553, 46777, 82505, 4290773, 4492205, 4976429, 21537833, 21549349, 51127261, 56786089, 60296573, 80837773, 87761789, 94424465, 138644873, 168865001, 221395541, 255881453, 297460453, 305198249, 360306365, 562654205, 635374253, 673867253, 808333573, 1164757553, 1210317349

Offset: 1

Max Alekseyev, Sep 13 2022

nonn

Also, odd integers n dividing 2^n + 8.

Some large terms: 5603900696716667005, 446661376165868432471569407934747098747181600670953926245, 1533278864164902082788937853692280620552397221686019535813.

Max Alekseyev, Table of n, a(n) for n = 1..781 (all terms below 10^15)

The odd terms of A245319.

Cf. A006521, A115976, A276967, A251603.

Select[Range[2155*10^4],PowerMod[2,#-3,#]==#-1&]//Quiet (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 08 2025 *)

A016057 Pseudo-powers to base 3: numbers k that are not powers of 3 such that k divides 2^k + 1.

Original entry on oeis.org

171, 513, 1539, 3249, 4617, 9747, 13203, 13851, 29241, 39609, 41553, 61731, 87723, 97641, 118827, 124659, 185193, 250857, 263169, 292923, 354537, 356481, 373977, 555579, 752571, 789507, 878769, 1063611, 1069443, 1121931, 1172889, 1666737, 1855179, 2152089, 2257713

Offset: 1

Robert G. Wilson v

nonn

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

Subsequence of A006521.

Do[ If[ PowerMod[ 2, n, n ] + 1 == n, If[ ! IntegerQ[ Log[ 3, n ] ], Print[ n ] ] ], {n, 1, 4* 10^6} ]

Offset corrected by Amiram Eldar, Jul 04 2021

A066807 Integers of the form (2^k+1)/k.

Original entry on oeis.org

3, 3, 57, 4971027, 29850020237398251227313, 17503832475167772961878049926333027042428946479619, 58167804601757508792865761341330650726830745576152064464908846133090363

Offset: 1

Benoit Cloitre, Jan 19 2002

nonn

The next term a(8) has 152 digits and a(9) has 217 digits. - Harvey P. Dale, Oct 20 2014

Amiram Eldar, Table of n, a(n) for n = 1..12

Cf. A006521.

Select[Table[(2^n+1)/n,{n,300}],IntegerQ] (* Harvey P. Dale, Oct 20 2014 *)

lista(nn) = {for (n=1, nn, if (type(v = (2^n+1)/n) == "t_INT", print1(v, ", ")););} \\ Michel Marcus, Nov 20 2013

a(n) == 0 (mod 3).

A069226 a(n) = gcd(n, 2^n + 1).

Original entry on oeis.org

2, 1, 1, 3, 1, 1, 1, 1, 1, 9, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 27, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 25, 3, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 17, 3, 5, 1, 1, 1, 1, 3, 1, 1, 13, 1, 1, 81, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1

Offset: 0

Vladeta Jovovic, Apr 12 2002

nonn

First occurrence of n: a(1)=1, a(3)=3, a(10)=5, a(9)=9, a(55)=11, a(78)=13, a(68)=17, a(50)=25, a(27)=27, a(406)=29, a(165)=33, a(666)=37, a(301)=43, a(1378)=53, a(1711)=59, a(390)=65, a(81)=81, a(3403)=83, a(2328)=97, a(495)=99, ... - R. J. Mathar, Dec 14 2016

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A006521 (fixed points), A014491, A049095, A049096, A060444.

Table[GCD[n,2^n+1],{n,100}] (* Harvey P. Dale, Dec 12 2012 *)

A069226(n) = gcd(n, 1+(1<Antti Karttunen, Jan 15 2025

Term a(0) = 2 prepended by Antti Karttunen, Jan 15 2025

A093547 Numbers k such that k divides 3^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 68, 80, 100, 110, 128, 136, 160, 164, 200, 220, 250, 256, 272, 320, 328, 340, 400, 440, 500, 512, 544, 550, 610, 640, 656, 680, 772, 800, 820, 880, 1000, 1010, 1024, 1088, 1100, 1156, 1210, 1220, 1250, 1280, 1312, 1360

Offset: 1

Farideh Firoozbakht, Mar 31 2004

nonn

This sequence is closed under multiplication, i.e., if x and y are terms then so is x*y.

A067945 is a subsequence of this sequence. A067945 is also closed under multiplication. In fact if m is an integer and k is a natural number then the sequence " n divides m^(n^k) - 1 " is a subsequence of the sequence " n divides m^n^(k+1)- 1 " and both are closed under multiplication.

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

Cf. A067945, A093546, A006521.

v={};Do[If[IntegerQ[(3^n^2-1)/n], v=Append[v, n];Print[v]], {n, 2500}]
Select[Range[1400],Divisible[3^#^2-1,#]&] (* Harvey P. Dale, Nov 04 2015 *)

isok(k) = Mod(3, k)^(k^2) == 1; \\ Amiram Eldar, May 26 2024

A327468 Numbers m that divide 8^m + 7.

Original entry on oeis.org

1, 3, 5, 25, 519, 290502305, 821808425, 979288025, 982989263, 25783323897, 27771237541, 31045665345, 65130752425, 3708883906025, 15079242289703, 973336048301405

Offset: 1

Juri-Stepan Gerasimov, Oct 04 2019

Conjecture: For k > 1, k^m == 1-k (mod m) has an infinite number of positive solutions.
Integer m not divisible by 3 is a term if and only if 3m is a term of A240941. - Max Alekseyev, Feb 07 2024
Also terms 930486448009391617725 and 21036656390681764555645540794214294457925. - Giovanni Resta, Oct 04 2019
Other terms 71245661271703622047, 7093208961478946798805, 7807963392818324067361574236385. - Max Alekseyev, Feb 07 2024

Solutions to k^m == 1-k (mod m): 1 (k = 1), A006521 (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), A328033 (k = 7), this sequence (k = 8).

Cf. A240941, A253211.

[m: m in [1..7] | (8^m + 7) mod m eq 0] cat [m: m in [8..10^8] | Modexp(8, m, m) + 7 eq m]; // Jon E. Schoenfield, Oct 05 2019

isok(n) = Mod(8, n)^n==-7; \\ Michel Marcus, Oct 05 2019

a(10)-a(13) from Giovanni Resta, Oct 04 2019

a(14)-a(16) from Max Alekseyev, Feb 07 2024

A093665 Numbers k that divide 2^(k^3) + 1.

Original entry on oeis.org

1, 3, 9, 27, 57, 81, 171, 243, 513, 729, 1083, 1467, 1539, 2187, 3249, 4401, 4617, 6561, 9747, 13131, 13203, 13851, 19683, 20577, 27873, 29241, 32547, 39393, 39609, 41553, 59049, 61731, 83619, 87723, 97641, 118179, 118827, 124659, 177147, 185193

Offset: 1

Robert G. Wilson v, Apr 03 2004

nonn

Cf. A006521, A093546.

Select[ Range[ 249488], PowerMod[2, #^3, # ] == # - 1 &]

A096197 a(n) = (1+prime(n)) mod n.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 4, 4, 6, 0, 10, 2, 3, 2, 3, 6, 9, 8, 11, 12, 11, 14, 15, 18, 23, 24, 23, 24, 23, 24, 4, 4, 6, 4, 10, 8, 10, 12, 12, 14, 16, 14, 20, 18, 18, 16, 24, 32, 32, 30, 30, 32, 30, 36, 38, 40, 42, 40, 42, 42, 40, 46, 56, 56, 54, 54, 64, 66, 3, 0, 70, 0, 3, 4, 5, 4, 5, 8, 7, 10, 15

Offset: 1

Labos Elemer, Jul 26 2004

Graph is similar to that of A004648.

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

Cf. A082495, A006521, A087611, A004648.

[(NthPrime(n)+1) mod(n): n in [1..90]]; // Vincenzo Librandi, Sep 11 2014

Table[Mod[Prime[n] + 1, n], {n, 100}] (* Vincenzo Librandi, Sep 11 2014 *)

lista(nn) = {forprime(p=2, n, print1((p+1) % primepi(p), ", "););} \\ Michel Marcus, Sep 11 2014

A289258 Prime factors of numbers in A289257.

Original entry on oeis.org

3, 19, 163, 1459, 8803, 17497, 52489, 78787, 164617, 370387

Offset: 1

Michel Marcus, Jun 29 2017

Kalmynin conjectures that this sequence is infinite.

I have some doubts about terms 17497, 52489, 164617 that are == 1 (mod 8).

Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See Theorem 7 p. 11.

Cf. A006521, A057719, A124240, A289257.

cp's OEIS Frontend

A328033 Numbers m that divide 7^m + 6.

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A357125 Positive integers n such that 2^(n-3) == -1 (mod n).

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A016057 Pseudo-powers to base 3: numbers k that are not powers of 3 such that k divides 2^k + 1.

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Mathematica

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A066807 Integers of the form (2^k+1)/k.

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Formula

A069226 a(n) = gcd(n, 2^n + 1).

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A093547 Numbers k such that k divides 3^(k^2) - 1.

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A327468 Numbers m that divide 8^m + 7.

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Magma

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Extensions

A093665 Numbers k that divide 2^(k^3) + 1.

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Mathematica

A096197 a(n) = (1+prime(n)) mod n.

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