A067945 -id:A067945 - cp's OEIS frontend

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

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

A277288 Positive integers k such that k divides 3^k + 5.

Original entry on oeis.org

1, 2, 14, 1978, 38209, 4782974, 9581014, 244330711, 365496202, 1661392258, 116084432414, 288504187458218, 490179448388654, 802245996685561

Offset: 1

Seiichi Manyama, Oct 09 2016

No other terms below 10^15. Some larger terms: 79854828136468902206, 3518556634988844968631084847788071912030455376274045370172567094578. - Max Alekseyev, Oct 14 2016

			3^14 + 5 = 4782974 = 14 * 341641, so 14 is a term.

Solutions to 3^n == k (mod n): A277340 (k=-11), A277289 (k=-7), this sequence (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==-5; \\ Joerg Arndt, Oct 09 2016

A277288_list = [1,2]+[n for n in range(3,10**6) if pow(3,n,n)==n-5] # Chai Wah Wu, Oct 09 2016

def A277288_list(search_limit):
    n, t, r = 1, Integer(3), [1]
    while n < search_limit:
        n += 1
        t *= 3
        if n.divides(t+5): r.append(n)
    return r # Peter Luschny, Oct 10 2016

a(9) from Joerg Arndt, Oct 09 2016
a(10) from Chai Wah Wu, Oct 09 2016
a(11)-a(14) from Max Alekseyev, Oct 14 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

A128357 Quotients A128356(n)/prime(n).

Original entry on oeis.org

10, 7, 311, 127, 23, 157, 343927, 7805561, 47, 9629, 311, 25679, 821, 1470086279, 12409, 71233, 1232333, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843, 1880415721, 263

Offset: 1

Alexander Adamchuk, Mar 02 2007, Mar 09 2007

A128356 = {20, 21, 1555, 889, 253, 2041, 5846759, ...} = Least number k>1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n). Most listed terms are primes, except a(7) = 20231*17 and a(8) = 410819*19. a(15) = 12409. a(16) = 71233.

Note that all prime listed terms of {a(n)} coincide with terms of A128456 = {2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, ...} = least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n).

Cf. A128356 (least number k > 1 (that is not a power of prime p) such that k divides (p+1)^k-1, where p = prime(n)).
Cf. A014960, A128360, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945.
Cf. A128456 (least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n)).

Terms a(14) onwards from Max Alekseyev, Feb 08 2010

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

A177805 Numbers k such that k divides 15^k - 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 136, 196, 224, 256, 272, 343, 392, 448, 452, 512, 544, 686, 784, 812, 896, 904, 952, 1024, 1088, 1372, 1568, 1624, 1792, 1808, 1904, 2048, 2176, 2312, 2401, 2744, 3136, 3164, 3248, 3584, 3616, 3808, 4096

Offset: 1

Alexander Adamchuk, May 17 2010

nonn

A000420 are the only odd terms of the sequence. - Robert Israel, Feb 25 2020

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000420, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128358, A128360.

Contains A003591.

filter:= n -> 15 &^ n - 1 mod n = 0:
select(filter, [$1..10000]); # Robert Israel, Feb 25 2020

is(n)=Mod(15,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

A093546 Numbers n such that n divides 2^n^2 + 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, 13203, 13851, 19683, 20577, 27873, 29241, 32547, 39393, 39609, 41553, 59049, 61731, 83619, 87723, 97641, 118179, 118827, 124659, 177147, 185193, 239121

Offset: 1

Farideh Firoozbakht, Mar 31 2004

nonn

This sequence is closed under multiplication. A006521 is a subsequence of this sequence. A006521 is also closed under multiplication. In fact if m is even 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.

"Closed under multiplication" means that if x and y are terms then so is x*y.

Charles R Greathouse IV, Table of n, a(n) for n = 1..415

Cf. A006521, A067945, A093547.

Select[ Range[250857], PowerMod[2, #^2, # ] == # - 1 &] (* Robert G. Wilson v, Apr 02 2004 *)

is(n)=Mod(2,n)^n^2==-1 \\ Charles R Greathouse IV, Aug 01 2016

Corrected and extended by Robert G. Wilson v, Apr 02 2004

A177807 Numbers k that divide 17^k - 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 42, 48, 54, 60, 64, 72, 78, 80, 84, 96, 100, 108, 116, 120, 126, 128, 144, 156, 160, 162, 168, 180, 192, 200, 216, 220, 232, 234, 240, 252, 256, 288, 294, 300, 312, 320, 324, 336, 342, 348, 360, 378, 384, 400, 420

Offset: 1

Alexander Adamchuk, May 17 2010

nonn

Zizheng Fang, Table of n, a(n) for n = 1..10000
Zizheng Fang, Python program to generate A177807

Cf. A014960, A014956, A014957, A014951, A014949, A014946, A014945, A067945, A128358, A128360, A177805.

{1}~Join~Select[Range[420], PowerMod[17, #, #] == 1 &] (* Giovanni Resta, Jan 30 2020 *)

cp's OEIS Frontend

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

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

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A277288 Positive integers k such that k divides 3^k + 5.

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

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A128357 Quotients A128356(n)/prime(n).

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

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

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A177805 Numbers k such that k divides 15^k - 1.

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