A295532
Positive integers n such that 13^n == 8 (mod n).
Original entry on oeis.org
1, 5, 371285, 3957661, 70348567451, 42831939409247
Offset: 1
Sequences 13^n == k (mod n):
A116621 (k=1),
A116622 (k=2),
A116629 (k=3),
A116630 (k=4),
A116611 (k=5),
A116631 (k=6),
A116632 (k=7), this sequence (k=8),
A116636 (k=9),
A116620 (k=10),
A116638 (k=11),
A116639 (k=15).
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Join[{1, 5}, Select[Range[4000000], PowerMod[13, #, #] == 8 &]] (* Robert Price, Apr 10 2020 *)
A277401
Positive integers n such that 7^n == 2 (mod n).
Original entry on oeis.org
1, 5, 143, 1133, 2171, 8567, 16805, 208091, 1887043, 517295383, 878436591673
Offset: 1
7 == 2 mod 1, so 1 is a term;
16807 == 2 mod 5, so 5 is a term.
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Join[{1},Select[Range[5173*10^5],PowerMod[7,#,#]==2&]] (* The program will generate the first 10 terms of the sequence; it would take a very long time to generate the 11th term. *) (* Harvey P. Dale, Apr 15 2020 *)
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isok(n) = Mod(7, n)^n == 2; \\ Michel Marcus, Oct 13 2016
A333269
Positive integers n such that 17^n == 2 (mod n).
Original entry on oeis.org
1, 3, 5, 3585, 4911, 5709, 1688565, 7361691, 16747709, 3226850283899, 8814126944005, 33226030397603
Offset: 1
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for(k=1, 1e6, if(Mod(17, k)^k==2, print1(k", ")))
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A333269_list = [n for n in range(1,10**6) if n == 1 or pow(17,n,n) == 2] # Chai Wah Wu, Mar 14 2020
A321364
Positive integers m such that 13^m == 12 (mod m).
Original entry on oeis.org
1, 13757837, 6969969233, 514208575135
Offset: 1
Solutions to 13^m == k (mod m):
A001022 (k=0),
A015963 (k=-1),
A116621 (k=1),
A116622 (k=2),
A116629 (k=3),
A116630 (k=4),
A116611 (k=5),
A116631 (k=6),
A116632 (k=7),
A295532 (k=8),
A116636 (k=9),
A116620(k=10),
A116638 (k=11), this sequence (k=12),
A321365 (k=14),
A116639 (k=15).
A321365
Positive integers n such that 13^n == 14 (mod n).
Original entry on oeis.org
1, 5805311, 392908759, 399614833907, 2674764845549, 21997277871211, 67146783889057
Offset: 1
Solutions to 13^n == k (mod n):
A001022 (k=0),
A015963 (k=-1),
A116621 (k=1),
A116622 (k=2),
A116629 (k=3),
A116630 (k=4),
A116611 (k=5),
A116631 (k=6),
A116632 (k=7),
A295532 (k=8),
A116636 (k=9),
A116620(k=10),
A116638 (k=11),
A321364 (k=12), this sequence (k=14),
A116639 (k=15).
A333134
Positive integers k such that 11^k == 2 (mod k).
Original entry on oeis.org
1, 3, 413, 1329, 6587, 11629, 75761, 925071199, 9031140861789, 114876097917387, 1314252479257933
Offset: 1
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for(k=1, 1e6, if(Mod(11, k)^k==2, print1(k", ")))
A333413
Positive integers k such that k divides 13^k + 2.
Original entry on oeis.org
1, 3, 5, 185, 2199, 14061, 5672119, 6719547, 192178873, 913591893, 4589621727, 9762178659, 1157052555699
Offset: 1
Solutions to 13^k == m (mod k): this sequence (m = -2),
A015963 (m = -1),
A116621 (m = 1),
A116622 (m = 2),
A116629 (m = 3),
A116630 (m = 4),
A116611 (m = 5),
A116631 (m = 6),
A116632 (m = 7),
A295532 (m = 8),
A116636 (m = 9),
A116620 (m = 10),
A116638 (m = 11),
A116639 (k = 15).
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Select[Range[100000], Divisible[PowerMod[13, #, #] + 2, #] &] (* Jinyuan Wang, Mar 28 2020 *)
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for(k=1, 1e6, if(Mod(13, k)^k==-2, print1(k", ")))
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