A015924 -id:A015924 - cp's OEIS frontend

A033984 Odd integers n such that 2^n == 16 (mod n).

1, 7, 40369, 673663, 990409, 1697609, 2073127, 6462649, 7527199, 7559479, 14421169, 21484129, 37825753, 57233047, 130647919, 141735559, 179203369, 188967289, 218206489, 259195009, 264538057, 277628449, 330662479, 398321239, 501126487

Offset: 1

Joe K. Crump (joecr(AT)carolina.rr.com)

nonn

The odd terms of A015924.

For all m, 2^A128121(m)-1 belongs to this sequence.

Max Alekseyev, Table of n, a(n) for n = 1..790 (all terms below 10^14)

Besides initial terms, the sequence coincides with A173138.

Cf. A015924, A173138, A173572.

Select[Range[1,510000001,2],PowerMod[2,#,#]==16&] (* Harvey P. Dale, Dec 11 2010 *)

Edited and terms 1,7 prepended by Max Alekseyev, Aug 09 2012

A128122 Numbers m such that 2^m == 6 (mod m).

Original entry on oeis.org

1, 2, 10669, 6611474, 43070220513807782

Offset: 1

Alexander Adamchuk, Feb 15 2007

No other terms below 10^17. - Max Alekseyev, Nov 18 2022

A large term: 862*(2^861-3)/281437921287063162726198552345362315020202285185118249390789 (203 digits). - Max Alekseyev, Sep 24 2016

			2 == 6 (mod 1), so 1 is a term;
4 == 6 (mod 2), so 2 is a term.

Joe K. Crump, 2^n mod n
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A000079 (k=0),A187787 (k=1/2), A296369 (k=-1/2), A006521 (k=-1), A296370 (k=3/2), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), this sequence (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Cf. A015910, A036236.

m = 6; Join[Select[Range[m], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 08 2018 *)

1 and 2 added by N. J. A. Sloane, Apr 23 2007

a(5) from Max Alekseyev, Nov 18 2022

A296369 Numbers m such that 2^m == -1/2 (mod m).

Original entry on oeis.org

1, 5, 65, 377, 1189, 1469, 25805, 58589, 134945, 137345, 170585, 272609, 285389, 420209, 538733, 592409, 618449, 680705, 778805, 1163065, 1520441, 1700945, 2099201, 2831009, 4020029, 4174169, 4516109, 5059889, 5215769

Offset: 1

Max Alekseyev, Dec 10 2017

nonn

Equivalently, 2^(m+1) == -1 (mod m), or m divides 2^(m+1) + 1.

The sequence is infinite, see A055685.

Chai Wah Wu, Table of n, a(n) for n = 1..1192
OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): A296370 (k=3/2), A187787 (k=1/2), this sequence (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Select[Range[10^5], Divisible[2^(# + 1) + 1, #] &] (* Robert Price, Oct 11 2018 *)

A296369_list = [n for n in range(1,10**6) if pow(2,n+1,n) == n-1] # Chai Wah Wu, Nov 04 2019

a(n) = A055685(n) - 1.

Incorrect term 4285389 removed by Chai Wah Wu, Nov 04 2019

A015926 Positive integers n such that 2^n == 2^6 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 24, 30, 31, 32, 36, 42, 48, 64, 66, 72, 78, 84, 90, 96, 102, 114, 126, 138, 144, 168, 174, 176, 186, 192, 210, 222, 234, 246, 252, 258, 282, 288, 318, 336, 354, 366, 390, 396, 402, 426, 438, 456, 474, 496, 498, 504, 510, 534, 546

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215610.

For all m, 2^A033981(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Contains A208154 as a subsequence.

Cf. A015919, A015921, A015922, A015924, A015925, A015927, A015929, A015931, A015932, A015935, A015937.

Select[Range[1000], Mod[2^# - 2^6, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)

Edited by Max Alekseyev, Jul 30 2011

A015929 Positive integers n such that 2^n == 2^8 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 32, 40, 48, 56, 60, 64, 80, 88, 96, 104, 120, 127, 128, 136, 140, 152, 160, 184, 192, 224, 232, 240, 248, 256, 260, 272, 296, 308, 320, 328, 344, 376, 384, 408, 416, 424, 472, 480

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215611.

For all m, 2^A051447(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Contains A208156 as a subsequence.

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015931, A015932, A015935, A015937.

Select[Range[1000], Mod[2^# - 2^8, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)

Edited by Max Alekseyev, Jul 30 2011

A015932 Positive integers n such that 2^n == 2^10 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 12, 16, 24, 28, 30, 32, 34, 48, 50, 64, 70, 73, 96, 110, 112, 128, 130, 150, 170, 190, 192, 230, 256, 290, 310, 330, 370, 384, 410, 430, 442, 448, 470, 512, 530, 532, 550, 590, 610, 670, 710

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215612.

For all m, 2^A033982(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Contains A208158 as a subsequence.

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015929, A015931, A015935, A015937.

Select[Range[1000], Mod[2^# - 2^10, #] == 0 &] (* T. D. Noe, Aug 17 2012 *)

Edited by Max Alekseyev, Jul 30 2011

A015937 Positive integers n such that 2^n == 2^12 (mod n).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 22, 23, 24, 32, 36, 40, 42, 48, 60, 62, 64, 68, 72, 80, 84, 89, 96, 120, 126, 128, 132, 144, 156, 160, 168, 180, 192, 204, 228, 240, 252, 256, 276, 288, 312, 320, 336, 340, 348, 352, 360, 372, 384, 420, 444, 462

Offset: 1

Robert G. Wilson v

nonn

The odd terms are given by A215613.

For all m, 2^A051446(m)-1 belongs to this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, 2^n mod n

Cf. A015919, A015921, A015922, A015924, A015925, A015926, A015927, A015929, A015931, A015932, A015935.

With[{c=2^12},Select[Range[1,6000],Divisible[2^#-c,#]&]] (* Harvey P. Dale, Mar 20 2011 *)

Edited by Max Alekseyev, Jul 31 2011

A173138 Composite numbers k such that 2^(k-4) == 1 (mod k).

Original entry on oeis.org

4, 40369, 673663, 990409, 1697609, 2073127, 6462649, 7527199, 7559479, 14421169, 21484129, 37825753, 57233047, 130647919, 141735559, 179203369, 188967289, 218206489, 259195009, 264538057, 277628449, 330662479, 398321239, 501126487, 506958313, 612368311, 767983759

Offset: 1

Michel Lagneau, Feb 10 2010

nonn

Besides the initial term, the sequence coincides with A033984 and consists of the odd terms > 7 of A015924.

			4 is a term: 2^(4 - 4) = 1 (mod 4).

A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

Michael S. Branicky, Table of n, a(n) for n = 1..62

Cf. A002808, A005381, A033984.

with(numtheory): for n from 1 to 100000000 do: a:= 2^(n-4)- 1; b:= a / n; c:= floor(b): if b = c and tau(n) <> 2 then print (n); else fi;od:

Select[Range[500000000],!PrimeQ[#]&&PowerMod[2,#-4,#]==1&] (* Harvey P. Dale, Nov 23 2011 *)

is(n)=!isprime(n) && n>1 && Mod(2,n)^(n-4)==1 \\ Charles R Greathouse IV, Nov 23 2011

from sympy import isprime, prime, nextprime
def afind(k=4):
    while True:
        if pow(2, k-4, k) == 1 and not isprime(k): print(k, end=", ")
        k += 1
afind() # Michael S. Branicky, Mar 21 2022

Simplified the definition, added cross-reference to A033984 R. J. Mathar, May 18 2010
More terms from Harvey P. Dale, Nov 23 2011
Typo in a(13) corrected by Georg Fischer, Mar 19 2022
a(24) and beyond from Michael S. Branicky, Mar 21 2022

A296370 Numbers m such that 2^m == 3/2 (mod m).

Original entry on oeis.org

1, 111481, 465793, 79036177, 1781269903307, 250369632905747, 708229497085909, 15673900819204067

Offset: 1

Max Alekseyev, Dec 11 2017

Equivalently, 2^(m+1) == 3 (mod m).
Also, numbers m such that 2^(m+1) - 2 is a Fermat pseudoprime base 2, i.e., 2^(m+1) - 2 belongs to A015919 and A006935.
Some larger terms (may be not in order): 2338990834231272653581, 341569682872976768698011746141903924998969680637.

OEIS Wiki, 2^n mod n

Solutions to 2^m == k (mod m): this sequence (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12)

Select[Range[10^6], Divisible[2^(# + 1) - 3, #] &] (* Robert Price, Oct 11 2018 *)

a(n) = A296104(n) - 1.

A334634 Numbers m that divide 2^m + 11.

Original entry on oeis.org

1, 13, 16043199041, 91118493923, 28047837698634913

Offset: 1

Max Alekseyev, Sep 10 2020

Equivalently, numbers m such that 2^m == -11 (mod m).

No other terms below 10^17.

OEIS Wiki, 2^n mod n

Solutions to 2^n == k (mod n): A296370 (k=3/2), A187787 (k=1/2), A296369 (k=-1/2), A000079 (k=0), A006521 (k=-1), A015919 (k=2), A006517 (k=-2), A050259 (k=3), A015940 (k=-3), A015921 (k=4), A244673 (k=-4), A128121 (k=5), A245318 (k=-5), A128122 (k=6), A245728 (k=-6), A033981 (k=7), A240941 (k=-7), A015922 (k=8), A245319 (k=-8), A051447 (k=9), A240942 (k=-9), A128123 (k=10), A245594 (k=-10), A033982 (k=11), this sequence (k=-11), A128124 (k=12), A051446 (k=13), A128125 (k=14), A033983 (k=15), A015924 (k=16), A124974 (k=17), A128126 (k=18), A125000 (k=19), A015925 (k=2^5), A015926 (k=2^6), A015927 (k=2^7), A015929 (k=2^8), A015931 (k=2^9), A015932 (k=2^10), A015935 (k=2^11), A015937 (k=2^12).

a(5) from Sergey Paramonov, Oct 10 2021

cp's OEIS Frontend

A033984 Odd integers n such that 2^n == 16 (mod n).

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A128122 Numbers m such that 2^m == 6 (mod m).

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A296369 Numbers m such that 2^m == -1/2 (mod m).

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A015926 Positive integers n such that 2^n == 2^6 (mod n).

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A015929 Positive integers n such that 2^n == 2^8 (mod n).

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A015932 Positive integers n such that 2^n == 2^10 (mod n).

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A015937 Positive integers n such that 2^n == 2^12 (mod n).

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A173138 Composite numbers k such that 2^(k-4) == 1 (mod k).

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