A277288 -id:A277288 - cp's OEIS frontend

A015973 Positive integers n such that n | (3^n + 2).

1, 5, 77, 278377, 3697489, 219596687717, 56865169816619

Offset: 1

No other terms below 10^15. Some larger term: 3142423971953435020522506484187. - Max Alekseyev, Aug 04 2011

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), this sequence (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277126 (k=7), A277274 (k=11).

a(1)=1 prepended and a(6)-a(7) added by Max Alekseyev, Aug 04 2011

A276740 Numbers n such that 3^n == 5 (mod n).

Original entry on oeis.org

1, 2, 4, 76, 418, 1102, 4687, 7637, 139183, 2543923, 1614895738, 9083990938, 23149317409, 497240757797, 4447730232523, 16000967516764, 65262766108619, 141644055557882

Offset: 1

Dmitry Ezhov, Sep 16 2016

No other terms below 10^15. Some larger terms: 194995887252090239, 2185052151122686482926861593785262. - Max Alekseyev, Oct 13 2016

			3 == 5 (mod 1), so 1 is a term;
9 == 5 (mod 2), so 2 is a term.

Cf. A066601.

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), this sequence (k=5), A277628 (k=6), A277126 (k=7), A277630 (k=8), A277274 (k=11).

Select[Range[10^7], PowerMod[3, #, #] == Mod[5, #] &] (* Michael De Vlieger, Sep 26 2016 *)

isok(n) = Mod(3, n)^n == Mod(5, n); \\ Michel Marcus, Sep 17 2016

A276740_list = [1,2,4]+[n for n in range(5,10**6) if pow(3,n,n) == 5] # Chai Wah Wu, Oct 04 2016

a(11)-a(13) from Chai Wah Wu, Oct 05 2016
a(14) from Lars Blomberg, Oct 12 2016
a(15)-a(18) from Max Alekseyev, Oct 13 2016
a(12) was missing Robert G. Wilson v, Oct 19 2016

A277126 Positive integers n such that 3^n == 7 (mod n).

Original entry on oeis.org

1, 2, 295, 883438, 252027511, 7469046275, 26782373099, 53191768475, 55246802458, 819613658855, 893727887879978

Offset: 1

Seiichi Manyama, Oct 06 2016

No other terms below 10^15. A larger term: 9135884036634915191945452485106476242. - Max Alekseyev, Oct 12 2016

Terms are not divisible by 127 (Alekseyev 2016).

			3 == 7 mod 1, so 1 is a term;
9 == 7 mod 2, so 2 is a term.

M. A. Alekseyev. "Problem 4101". Crux Mathematicorum 42:1 (2016), 28.

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), this sequence (k=7), A277274 (k=11).

isok(n) = Mod(3, n)^n == 7; \\ Michel Marcus, Oct 06 2016

a(5) from Joerg Arndt, Oct 06 2016

a(6)-a(11) from Max Alekseyev, Oct 12 2016

A277289 Positive integers n such that n | (3^n + 7).

Original entry on oeis.org

1, 2, 4, 5, 8, 25, 44, 4664, 6568, 1353025, 2919526, 5709589, 7827725, 64661225, 85132756, 153872408, 743947534, 34304296003, 38832409867, 40263727492, 1946603375348, 2469908330348, 64471909888247, 274267749806485, 888906849689897, 896501949422459

Offset: 1

Seiichi Manyama, Oct 09 2016

nonn

No other terms below 10^15. - Max Alekseyev, Oct 14 2016

492385451091805616444 is a term.

			3^25 + 7 = 847288609450 = 25 * 33891544378, so 25 is a term.

Solutions to 3^n == k (mod n): A277340 (k=-11), this sequence (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277126 (k=7), A277274 (k=11).

is(n)=Mod(3,n)^n==-7; \\ Joerg Arndt, Oct 09 2016

A277289_list = [1,2,4,5]+[n for n in range(6,10**6) if pow(3,n,n)==n-7] # Chai Wah Wu, Oct 12 2016

a(17) from Joerg Arndt, Oct 09 2016
a(18)-a(20) from Chai Wah Wu, Oct 12 2016
a(21)-a(26) from Max Alekseyev, Oct 14 2016

A277274 Positive integers n such that 3^n == 11 (mod n).

Original entry on oeis.org

1, 2, 1162, 1692934, 3851999, 274422823, 14543645261, 492230729674, 773046873382, 13010754158393, 31446154470014, 583396812890467, 598371102650063

Offset: 1

Seiichi Manyama, Oct 08 2016

No other terms below 10^15. Some larger terms: 38726095838775708310162, 2682806839696008709567739369. - Max Alekseyev, Oct 12 2016

			3 == 11 mod 1, so 1 is a term.
9 == 11 mod 2, so 2 is a term.

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277126 (k=7), this sequence (k=11).

k = 3; lst = {1, 2}; While[k < 12000000001, If[ PowerMod[3, k, k] == 11, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Oct 08 2016 *)

a(7)-a(13) from Max Alekseyev, Oct 12 2016

A277340 Positive integers n such that n | (3^n + 11).

Original entry on oeis.org

1, 2, 4, 7, 10, 92, 1099, 29530, 281473, 657892, 3313964, 9816013, 18669155396, 94849225930, 358676424226, 957439868543, 1586504109310, 41431374800470, 241469610359708, 256165266592379

Offset: 1

Seiichi Manyama, Oct 09 2016

nonn

No other terms below 10^15. Some larger terms: 9151612250553176993, 1401778935853533028413047652833, 5645122353966835994338815444821661584288016927879134, 313*(3^626+11)/6562567821545333606830 (280 digits). - Max Alekseyev, Oct 14 2016

			3^10 + 11 = 59060 = 10 * 5906, so 10 is a term.

Solutions to 3^n == k (mod n): this sequence (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277126 (k=7), A277274 (k=11).

is(n)=Mod(3,n)^n==-11; \\ Joerg Arndt, Oct 10 2016

A277340_list = [1,2,4,7,10]+[n for n in range(11,10**6) if pow(3,n,n)==n-11] # Chai Wah Wu, Oct 11 2016

a(13)-a(14) from Chai Wah Wu, Oct 12 2016

a(15)-a(20) from Max Alekseyev, Oct 14 2016

A327943 Numbers m that divide 6^m + 5.

Original entry on oeis.org

1, 11, 341, 186787, 8607491, 9791567, 11703131, 14320387, 50168819, 952168003, 71654478989, 1328490399527

Offset: 1

Juri-Stepan Gerasimov, Sep 30 2019

Conjecture: For k > 1, k^m == 1 - k (mod m) has infinitely many positive solutions.
Also includes 11834972807906571233 = 31*381773316384082943. - Robert Israel, Oct 03 2019
a(13) > 10^15. - Max Alekseyev, Nov 10 2022

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

Cf. A245318, A277288, A015891, A253209.

[1] cat [n: n in [1..10^8] | Modexp(6, n, n) + 5 eq n];

Join[{1},Select[Range[98*10^5],PowerMod[6,#,#]==#-5&]] (* The program generates the first six terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Feb 05 2022 *)

a(11) from Giovanni Resta, Oct 02 2019

a(12) from Max Alekseyev, Nov 10 2022

A277628 Positive integers n such that 3^n == 6 (mod n).

Original entry on oeis.org

1, 3, 21, 936340943, 10460353197, 9374251222371, 23326283250291, 615790788171551

Offset: 1

Dmitry Ezhov, Oct 24 2016

No other terms below 10^15. - Max Alekseyev, Sep 12 2017

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), this sequence (k=6), A277126 (k=7), A277630 (k=8), A277274 (k=11).

PARI
```
isok(n) = Mod(3, n)^n == Mod(6, n);
```

a(6)-a(8) from Max Alekseyev, Sep 12 2017

A277630 Positive integers n such that 3^n == 8 (mod n).

Original entry on oeis.org

1, 5, 2352527, 193841707, 17126009179703, 380211619942943

Offset: 1

Dmitry Ezhov, Oct 24 2016

No other terms below 10^15. - Max Alekseyev, Sep 13 2017

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), A277288 (k=-5), A015973 (k=-2), A015949 (k=-1), A067945 (k=1), A276671 (k=2), A276740 (k=5), A277628 (k=6), A277126 (k=7), this sequence (k=8), A277274 (k=11).

PARI
```
isok(n) = Mod(3, n)^n == Mod(8, n);
```

a(5)-a(6) established by Max Alekseyev, Sep 13 2017

cp's OEIS Frontend

A015973 Positive integers n such that n | (3^n + 2).

Views

Author

Keywords

Comments

Crossrefs

Extensions

A276740 Numbers n such that 3^n == 5 (mod n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A277126 Positive integers n such that 3^n == 7 (mod n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions

A277289 Positive integers n such that n | (3^n + 7).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions

A277274 Positive integers n such that 3^n == 11 (mod n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A277340 Positive integers n such that n | (3^n + 11).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions

A327943 Numbers m that divide 6^m + 5.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Extensions

A277628 Positive integers n such that 3^n == 6 (mod n).

Views

Author

Keywords

Comments

Crossrefs

Programs