A015910 -id:A015910 - cp's OEIS frontend

A082494 a(n) = n - (2^n (mod n)).

1, 2, 1, 4, 3, 2, 5, 8, 1, 6, 9, 8, 11, 10, 7, 16, 15, 8, 17, 4, 13, 18, 21, 8, 18, 22, 1, 12, 27, 26, 29, 32, 25, 30, 17, 8, 35, 34, 31, 24, 39, 20, 41, 28, 28, 42, 45, 32, 19, 26, 43, 36, 51, 26, 12, 24, 49, 54, 57, 44, 59, 58, 55, 64, 33, 2, 65, 52, 61, 26, 69, 8, 71, 70, 7, 60, 59

Offset: 1

Anonymous, Apr 28 2003

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A015910, A082493, A215747.

a:= n-> n-(2&^n mod n):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2015

A112987 a(n) = 2^(2^n mod n) for n > 0; a(0) = 2.

Original entry on oeis.org

2, 1, 1, 4, 1, 4, 16, 4, 1, 256, 16, 4, 16, 4, 16, 256, 1, 4, 1024, 4, 65536, 256, 16, 4, 65536, 128, 16, 67108864, 65536, 4, 16, 4, 1, 256, 16, 262144, 268435456, 4, 16, 256, 65536, 4, 4194304, 4, 65536, 131072, 16, 4, 65536, 1073741824, 16777216, 256

Offset: 0

Paul Barry, Oct 08 2005

The definition of a(0) is motivated by the idea that (anything)^n = 1 for n = 0. We also get this if "mod n" is replaced by "in Z/nZ", for n = 0. - M. F. Hasler, Nov 09 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A015910.

[2] cat [2^Modexp(2, n, n): n in [1..60]]; // Vincenzo Librandi, Nov 09 2018

Join[{2}, 2^Table[PowerMod[2, n, n], {n, 85}]] (* Vincenzo Librandi, Nov 09 2018 *)

apply( A112987(n)=2^lift(if(n,Mod(2,n))^n), [0..50]) \\ M. F. Hasler, Nov 09 2018

a(n) = 2^A015910(n) for n > 0. [Corrected by M. F. Hasler, Nov 09 2018]

Name edited by M. F. Hasler, Nov 09 2018

A128092 a(n) = largest multiple of n which is <= 2^n.

Original entry on oeis.org

2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4092, 8190, 16380, 32760, 65536, 131070, 262134, 524286, 1048560, 2097144, 4194300, 8388606, 16777200, 33554425, 67108860, 134217702, 268435440, 536870910, 1073741820, 2147483646

Offset: 1

Leroy Quet, Feb 14 2007

nonn

Cf. A128093, A000799.

Cf. A000079, A015910.

a:=n->n*floor(2^n/n): seq(a(n),n=1..37); # Emeric Deutsch, Feb 16 2007

f[n_] := n*Floor[2^n/n];Array[f, 33] (* Ray Chandler, Feb 19 2007 *)

def A128092(n): return (m:=1<Chai Wah Wu, Aug 24 2023

a(n) = n*floor(2^n/n) = n*A000799(n).

a(n) = 2^n - (2^n mod n). - Chai Wah Wu, Aug 24 2023

Extended by Emeric Deutsch and Ray Chandler, Feb 19 2007

A215747 a(n) = (-2)^n mod n.

Original entry on oeis.org

0, 0, 1, 0, 3, 4, 5, 0, 1, 4, 9, 4, 11, 4, 7, 0, 15, 10, 17, 16, 13, 4, 21, 16, 18, 4, 1, 16, 27, 4, 29, 0, 25, 4, 17, 28, 35, 4, 31, 16, 39, 22, 41, 16, 28, 4, 45, 16, 19, 24, 43, 16, 51, 28, 12, 32, 49, 4, 57, 16, 59, 4, 55, 0, 33, 64, 65, 16, 61, 44, 69, 64, 71, 4, 7

Offset: 1

Alex Ratushnyak, Aug 23 2012

n^(n+2) mod (n+2) is essentially the same.
Indices of 0's: 2^k - 1, k>=0.
Indices of 1's: A006521 except the first term.
Indices of 3's: A015940.
Indices of 5's: 7, 133, 1517, 11761, ...
a(A000040(n)) = A000040(n)-2 = A040976(n).

			a(5) = (-2)^5 mod 5 = -32 mod 5 = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A015910, A082493, A082494, A110146, A213381.

a:= n-> (-2)&^n mod n:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2015

a[n_]:=Mod[(-2)^n ,n]; Array[a,75] (* Stefano Spezia, Aug 25 2025 *)

for n in range(1, 333):
    print((-2)**n % n, end=',')

A220235 a(n) = (2^n + 3^n) modulo n.

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 5, 1, 8, 3, 5, 1, 5, 13, 5, 1, 5, 1, 5, 17, 14, 13, 5, 1, 0, 13, 26, 13, 5, 13, 5, 1, 2, 13, 30, 1, 5, 13, 35, 17, 5, 37, 5, 9, 35, 13, 5, 1, 12, 23, 35, 45, 5, 1, 0, 41, 35, 13, 5, 37, 5, 13, 35, 1, 15, 1, 5, 29, 35, 13, 5, 1, 5, 13, 50

Offset: 1

Zak Seidov, Dec 08 2012

nonn

a(n) = (A015910(n) + A066601(n)) mod n.

a(n) = 0 at n = 1, 5, 25, 55, 125, 275, 605, 625, ... (A045576).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A015910 (2^n mod n), A066601 (3^n mod n), A045576 (n|(2^n + 3^n)).

Mathematica
```
Table[Mod[2^n + 3^n, n],{n,100}]
```

A294389 a(n) = 2^(n-3) mod n, for n >= 3.

Original entry on oeis.org

1, 2, 4, 2, 2, 0, 1, 8, 3, 8, 10, 4, 1, 0, 13, 8, 5, 12, 1, 6, 6, 8, 4, 20, 10, 16, 22, 8, 8, 0, 1, 26, 11, 8, 28, 10, 1, 32, 31, 8, 11, 24, 19, 12, 12, 32, 16, 28, 1, 28, 40, 44, 26, 32, 1, 44, 15, 32, 46, 16, 1, 0, 4, 8, 17, 36, 1, 58, 18, 8, 55, 56, 46, 40, 60, 8, 20, 32, 10, 62, 21, 8, 4, 22

Offset: 3

Enrique Navarrete, Oct 29 2017

nonn

Every nonnegative integer seems to appear in the sequence, and every integer seems to appear in the sequence of first differences (see link).

For all 3 <= n < 10^9, a(n) != 7. - Robert G. Wilson v, Nov 30 2017

G. C. Greubel, Table of n, a(n) for n = 3..10000
Enrique Navarrete, Sequences Derived from Residues of 2^n (mod n)

Cf. A015910, A212844, A213859.

[Modexp(2,(n-3),n): n in [3..100]]; // G. C. Greubel, Jan 11 2023

Array[PowerMod[2, # - 3, #] &, 80, 3] (* Robert G. Wilson v, Nov 30 2017 *)

[power_mod(2,(n-3),n) for n in range(3,101)] # G. C. Greubel, Jan 11 2023

More terms from Robert G. Wilson v, Nov 30 2017

A015948 a(n) = smallest k >= n such that k | (2^k + n).

Original entry on oeis.org

1, 2, 5, 4, 7, 10, 15, 8, 11, 14, 13, 28, 21, 78, 17, 16, 19, 22, 49, 42, 23, 26, 1577, 40, 33, 30, 29, 44, 31, 34, 39, 32, 65, 38, 37, 52, 115, 102, 41, 242, 43, 46, 51, 60, 47, 279, 395, 152, 57, 114, 53, 68, 85, 58, 63, 104, 59, 62, 61, 76, 69, 126, 5773, 64, 67, 1090

Offset: 0

Robert G. Wilson v

nonn

Equally, a(n) = smallest k with 2^k mod k = k - n.

Cf. A015910, A036236.

a(p-2) = p for p prime >= 5; a(2^k) = 2^k. - David W. Wilson

Edited by N. J. A. Sloane, Jan 31 2009 at the suggestion of R. J. Mathar and T. D. Noe.

Restricted the range of k in the definition - R. J. Mathar, Mar 07 2010

A073800 Remainder of division 2^n/c(n), where c(n)=A002808(n), the n-th composite.

Original entry on oeis.org

2, 4, 0, 7, 2, 4, 2, 1, 0, 16, 8, 1, 8, 16, 18, 16, 14, 8, 8, 0, 2, 30, 18, 28, 14, 4, 8, 16, 28, 19, 6, 16, 29, 34, 8, 40, 2, 14, 8, 16, 14, 4, 8, 4, 0, 49, 62, 52, 32, 4, 8, 46, 17, 20, 65, 22, 32, 16, 62, 64, 32, 64, 41, 16, 32, 64, 48, 70, 48, 24, 32, 22, 74, 84, 8, 16, 32, 52

Offset: 1

Labos Elemer, Aug 12 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A000079, A002808, A015910, A062173, A064367, A073797, A073798, A073799.

Table[Mod[2^j, FixedPoint[j+PrimePi[ # ]+1&, j]], {j, 1, 128}]
Module[{c=Select[Range[200],CompositeQ],len},len=Length[c];Table[ PowerMod[ 2,n,c[[n]]],{n,len}]] (* Harvey P. Dale, Mar 03 2018 *)

A180060 a(n) = 2^(2^n mod n) mod n.

Original entry on oeis.org

0, 1, 1, 1, 4, 4, 4, 1, 4, 6, 4, 4, 4, 2, 1, 1, 4, 16, 4, 16, 4, 16, 4, 16, 3, 16, 13, 16, 4, 16, 4, 1, 25, 16, 29, 16, 4, 16, 22, 16, 4, 16, 4, 20, 32, 16, 4, 16, 22, 16, 1, 16, 4, 52, 8, 32, 28, 16, 4, 16, 4, 16, 4, 1, 61, 16, 4, 52, 49, 46, 4, 16, 4, 16, 31, 24, 36, 16, 4, 16

Offset: 1

Juri-Stepan Gerasimov, Jan 14 2011

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000079, A179976.

Table[Mod[2^(Mod[2^n, n]), n], {n, 50}] (* Alonso del Arte, Jan 14 2011 *)
Table[PowerMod[2,PowerMod[2,n,n],n],{n,80}] (* Harvey P. Dale, Oct 09 2019 *)

a(n) = 2^A015910(n) mod n.

Entries corrected by R. J. Mathar, Jan 14 2011

A226221 Numbers n such that 2^n mod n is not a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 25, 27, 32, 35, 36, 42, 45, 49, 50, 54, 55, 64, 70, 75, 77, 81, 88, 91, 95, 98, 99, 100, 104, 105, 108, 110, 115, 117, 119, 121, 125, 128, 130, 135, 136, 140, 143, 147, 150, 152, 153, 155, 156, 160, 161, 162, 169, 171, 175, 180, 184, 187, 189, 190, 198, 200

Offset: 1

J. M. Bergot and Charles R Greathouse IV, May 31 2013

nonn

All terms beyond the first two are composite: this is a subsequence of A065090.

			2^18 = 262144 = 10 mod 18 and 10 is not a power of 2, so 18 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A015910, A015911.

isA226221 := proc(n)
    local m ;
    if n <= 2 then
        return true;
    end if;
    m := A015910(n) ;
    if type(m,'odd') or m = 0 then
        true;
    elif nops(numtheory[factorset](m))  >1 then
        true;
    else
        false;
    end if;
end proc:
A226221 := proc(n)
    local a;
    if n <= 2 then
        n;
    else
        for a from procname(n-1)+1 do
            if isA226221(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A226221(n),n=1..30) ; # R. J. Mathar, Jun 06 2013

Select[Range[200],!IntegerQ[Log[2,PowerMod[2,#,#]]]&] (* Harvey P. Dale, Feb 28 2022 *)

PARI
```
ispow2(n)=n>0 && n==1<
				
```

cp's OEIS Frontend

A082494 a(n) = n - (2^n (mod n)).

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Maple

A112987 a(n) = 2^(2^n mod n) for n > 0; a(0) = 2.

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Magma

Mathematica

PARI

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A128092 a(n) = largest multiple of n which is <= 2^n.

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Maple

Mathematica

Python

Formula

Extensions

A215747 a(n) = (-2)^n mod n.

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Maple

Mathematica

Python

A220235 a(n) = (2^n + 3^n) modulo n.

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Mathematica

A294389 a(n) = 2^(n-3) mod n, for n >= 3.

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Magma

Mathematica

SageMath

Extensions

A015948 a(n) = smallest k >= n such that k | (2^k + n).

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Keywords

Comments

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Formula

Extensions

A073800 Remainder of division 2^n/c(n), where c(n)=A002808(n), the n-th composite.

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