A062175 -id:A062175 - cp's OEIS frontend

A176997 Integers k such that 2^(k-1) == 1 (mod k).

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Juri-Stepan Gerasimov, Dec 08 2010

nonn

Old definition was: Odd integers n such that 2^(n-1) == 4^(n-1) == 8^(n-1) == ... == k^(n-1) (mod n), where k = A062383(n). Dividing 2^(n-1) == 4^(n-1) (mod n) by 2^(n-1), we get 1 == 2^(n-1) (mod n), implying the current definition. - Max Alekseyev, Sep 22 2016
The union of {1}, the odd primes, and the Fermat pseudoprimes, i.e., {1} U A065091 U A001567. Also, the union of A006005 and A001567 (conjectured by Alois P. Heinz, Dec 10 2010). - Max Alekseyev, Sep 22 2016
These numbers were called "fermatians" by Shanks (1962). - Amiram Eldar, Apr 21 2024

			5 is in the sequence because 2^(5-1) == 4^(5-1) == 8^(5-1) == 1 (mod 5).

Daniel Shanks, Solved and Unsolved Problems in Number Theory, Spartan Books, Washington D.C., 1962.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

The odd terms of A015919.
Odd integers n such that 2^n == 2^k (mod n): this sequence (k=1), A173572 (k=2), A276967 (k=3), A033984 (k=4), A276968 (k=5), A215610 (k=6), A276969 (k=7), A215611 (k=8), A276970 (k=9), A215612 (k=10), A276971 (k=11), A215613 (k=12).
Cf. A000079, A062173, A062175, A176817.

m = 1; Join[Select[Range[m], Divisible[2^(# - 1) - m, #] &],
Select[Range[m + 1, 10^3], PowerMod[2, # - 1, #] == m &]] (* Robert Price, Oct 12 2018 *)

isok(n) = Mod(2, n)^(n-1) == 1; \\ Michel Marcus, Sep 23 2016

from itertools import count, islice
def A176997_gen(startvalue=1): # generator of terms >= startvalue
    if startvalue <= 1:
        yield 1
    k = 1<<(s:=max(startvalue,1))-1
    for n in count(s):
        if k % n == 1:
            yield n
        k <<= 1
A176997_list = list(islice(A176997_gen(),30)) # Chai Wah Wu, Jun 30 2022

Edited by Max Alekseyev, Sep 22 2016

A062172 Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 1, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 0, 0, 1, 4, 3, 1, 5, 1, 3, 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 7, 5, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 4, 7, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 5, 0, 7, 1, 3, 0, 3, 0, 1, 0

Offset: 1

Henry Bottomley, Jun 12 2001

			T(5,3)=5^(3-1) mod 3=25 mod 3=1. Rows start (0,1,1,1,1,...), (0,0,1,0,1,...), (0,1,0,3,1...), (0,0,1,0,1,...), (0,1,1,1,0,...), ...

Cf. A002997, A060154. Rows include A057427, A062173, A062174, A062175, A062176. Columns include A000004, A000035, A011655, A010684 with interleaved 0's, A011558, A010875. Diagonals include all the rows again and A000004 and A009001 unsigned.

A176176 Numbers k such that 2^(k-1) == 4^(k-1) (mod k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 28, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 112, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167

Offset: 1

Juri-Stepan Gerasimov, Dec 07 2010

nonn

Numbers k such that A062173(k) = A062175(k).
Question: is the sequence (Powers of 2) UNION (odd primes), the union of A000079 and A005408?
The answer to the question is no: 2^(28-1) mod 28 = 4^(28-1) mod 28 = 8. Also, any base-2 Fermat pseudoprime (A001567) is a term of this sequence. - D. S. McNeil, Dec 07 2010

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000079, A005408, A062173.

Select[Range[200],PowerMod[2,#-1,#]==PowerMod[4,#-1,#]&] (* Harvey P. Dale, Nov 10 2011 *)

Extended by D. S. McNeil, Dec 07 2010

A176817 Nonprimes k such that 2^(k-1) == 4^(k-1) (mod k).

Original entry on oeis.org

1, 4, 8, 16, 28, 32, 64, 112, 128, 256, 341, 448, 496, 512, 561, 645, 1016, 1024, 1105, 1288, 1387, 1729, 1792, 1905, 2044, 2047, 2048, 2416, 2465, 2701, 2821, 3277, 4033, 4096, 4369, 4371, 4672, 4681, 4984, 5461, 6601, 7168, 7936, 7957, 8128, 8192, 8321, 8481, 8911

Offset: 1

Juri-Stepan Gerasimov, Dec 07 2010

nonn

All powers of two are present.

Cf. A062173, A062175, A176176.

 fQ[n_] := !PrimeQ@ n && PowerMod[2, n - 1, n] == PowerMod[4, n - 1, n]; Select[ Range@ 10000, fQ]

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A176997 Integers k such that 2^(k-1) == 1 (mod k).

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A062172 Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].

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A176176 Numbers k such that 2^(k-1) == 4^(k-1) (mod k).

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A176817 Nonprimes k such that 2^(k-1) == 4^(k-1) (mod k).

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