A033984 -id:A033984 - cp's OEIS frontend

A276971 Odd integers n such that 2^n == 2^11 (mod n).

1, 3, 11, 15, 31, 35, 51, 121, 341, 451, 455, 671, 781, 1111, 1235, 1271, 1441, 1547, 1661, 1991, 2091, 2101, 2321, 2651, 2761, 2981, 3091, 3421, 3641, 3731, 3751, 4403, 4411, 4631, 4741, 5071, 5401, 5731, 5951, 6171, 6191, 6281, 6611, 6851, 6941, 7051, 7271, 7601, 7711, 8261, 8371, 8435, 8921

Offset: 1

Max Alekseyev, Sep 22 2016

Also, integers n such that 2^(n-11) == 1 (mod n).

For all m, 2^A128124(m)-1 belongs to this sequence.

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

The odd terms of A015935.
Odd integers n such that 2^n == 2^k (mod n): A176997 (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), this sequence (k=11), A215613 (k=12).
Cf. A128124.

m = 2^11; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &],
Select[Range[m + 1, 10^6, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)

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

A242875 Numbers n such that (n^n+2^2)/(n+2) is an integer.

Original entry on oeis.org

2, 5, 8, 128, 2144, 4808, 12872, 14168, 33672, 40367, 45992, 116192, 185768, 186824, 271208, 426008, 524288, 601352, 612768, 673661, 755792, 990407, 996032, 1697607, 1878368, 2073125, 2262752, 4325960, 4810808, 6331808, 6462647, 6707352, 7527197, 7559477

Offset: 1

Derek Orr, May 25 2014

nonn

Given S = (n^n+k^k)/(n+k) (here k = 2), when k = 2^m for some m > 0, there are significantly less values of n that make S an integer. For k=3, see A242883.
a(15) > 210000.
Equivalently, (-2)^n + 4 == 0 (mod n + 2). - Robert Israel, Jun 10 2014
Odd terms are A033984(2..infinity) - 2. - Robert Israel, Jun 10 2014

			(5^5+2^2)/(5+2) = 3129/7 = 447 is an integer. Thus 5 is a member of this sequence.

Robert Israel, Table of n, a(n) for n = 1..100

Cf. A242883.

filter:= proc(n) (-2)&^n + 4 mod (n+2) = 0 end proc;
select(filter, [$1..10^6]); # Robert Israel, Jun 10 2014

for(n=1,10^5,s=(n^n+2^2)/(n+2);if(floor(s)==s,print(n)))

a(16)-a(34) from Robert Israel, Jun 10 2014

A294390 a(n) = 2^(n-4) mod n, for n >= 4.

Original entry on oeis.org

1, 2, 4, 1, 0, 5, 4, 7, 4, 5, 2, 8, 0, 15, 4, 12, 16, 11, 14, 3, 16, 2, 10, 5, 8, 11, 4, 4, 0, 17, 30, 23, 4, 14, 24, 20, 16, 36, 4, 27, 12, 32, 6, 6, 16, 8, 14, 26, 40, 20, 22, 13, 16, 29, 22, 37, 16, 23, 8, 32, 0, 2, 4, 42, 52, 35, 64, 9, 40, 64, 28, 23, 20, 30, 4

Offset: 4

Enrique Navarrete, Oct 29 2017

Every nonnegative integer seems to appear in the sequence, and every integer seems to appear in the sequence of first differences (see link).
From Robert Israel, Dec 04 2017: (Start)
a(n) = 0 iff n>=8 is a power of 2.
a(n) = 1 iff n=4 or n is in A033984.
a(n) = 2 iff n>=4 is in A015925 and is not divisible by 4. (End)

			For n=9, 2^5 = 32 == 5 mod 9.

Robert Israel, Table of n, a(n) for n = 4..10000
Enrique Navarrete, Sequences derived from residues of 2^n (mod n)

Cf. A015910, A015925, A033984, A294389.

[Modexp(2, n-4, n): n in [4..120]]; // G. C. Greubel, Dec 27 2024

A294390:=n->2&^(n-4) mod n: seq(A294390(n), n=4..150); # Wesley Ivan Hurt, Nov 30 2017

Array[Mod[2^(# - 4), #] &, 75, 4] (* Michael De Vlieger, Dec 02 2017 *)
Array[PowerMod[2,#-4,#]&,80,4] (* Harvey P. Dale, Dec 01 2018 *)

a(n) = lift(Mod(2, n)^(n-4)); \\ Michel Marcus, Oct 30 2017

print([power_mod(2, n-4, n) for n in range(4, 101)]) # G. C. Greubel, Dec 27 2024

More terms from Michel Marcus, Oct 30 2017

A385073 a(n) = b^(n-1) mod n, where b = A053669(n) is the least integer greater than 1 and coprime to n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 3, 4, 3, 1, 5, 1, 3, 4, 11, 1, 11, 1, 7, 4, 3, 1, 5, 16, 3, 13, 27, 1, 7, 1, 11, 4, 3, 9, 29, 1, 3, 4, 27, 1, 17, 1, 27, 31, 3, 1, 29, 15, 33, 4, 27, 1, 11, 49, 3, 4, 3, 1, 43, 1, 3, 4, 43, 16, 23, 1, 27, 4, 13, 1, 29, 1, 3, 34, 27, 9, 5, 1, 27, 40, 3, 1, 17

Offset: 1

Robert G. Wilson v, Jun 16 2025

Inspired by Fermat's Little Theorem.

a(n) > 0 for n > 1 since n and b are coprime.

Wikipedia, Fermat's little theorem

Cf. A053669, A385074.

f:= proc(n) local b;
  b:= 2;
  while n mod b = 0 do b:= nextprime(b) od;
  b &^ (n-1) mod n
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Jun 18 2025

a[n_] := Block[{b = 2}, While[GCD[n, b] > 1, b++]; PowerMod[b, n - 1, n]]; Array[a, 84]

a(n) = forprime(p=2, , if(n%p, return(lift(Mod(p, n)^(n-1))))); \\ Michel Marcus, Jun 18 2025

a(n) = 0 iff n = 1.
a(n) = 1 iff n belongs to A000040, A001567, or A130433.
a(n) = 2 iff n>1 and belongs to A173572;
a(n) = 4 iff n belongs to A033553;
a(n) = 8 iff n>7 and belongs to either A033984 or A173138;
a(n) = 16 iff n>15 and belongs to A276968;
a(n) = 32 iff n>1 and belongs to A215610;
a(n) = 64 iff n>63 and belongs to A276969;
a(n) = 128 iff n>127 and belongs to A215611;
a(n) = 256 iff n>255 and belongs to A276970;
a(n) = 512 iff n>511 and belongs to A215612;
a(n) = 1024 iff n>1023 and belongs to A276971;
a(n) = 2048 iff n>2047 and belongs to A215613;
From Robert Israel, Jun 18 2025: (Start)
a(2*p) = 3 if p is a prime > 3.
a(3*p) = 4 if p is a prime > 2.
a(4*p) = 3^3 if p is a prime > 5.
a(6*p) = 5^5 if p is a prime > 509.
a(8*p) = 3^5 if p is a prime > 271.
a(10*p) = 3^9 if p is a prime > 1951.
a(12*p) = 5^11 if p is a prime > 4069003. (End)

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A276971 Odd integers n such that 2^n == 2^11 (mod n).

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

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A242875 Numbers n such that (n^n+2^2)/(n+2) is an integer.

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A294390 a(n) = 2^(n-4) mod n, for n >= 4.

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A385073 a(n) = b^(n-1) mod n, where b = A053669(n) is the least integer greater than 1 and coprime to n.

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