A213381 -id:A213381 - cp's OEIS frontend

A213407 First occurrence of n in A213381.

2, 0, 3, 5, 4, 9, 8, 7, 12, 17, 594270126787, 21, 20, 13, 24, 29, 16, 23, 32, 19, 36, 41, 1356, 45, 38, 151, 51, 411, 994, 57, 56, 31, 60, 65, 52, 69, 71, 37, 72, 77, 490, 81, 80, 169, 84, 6125, 68, 1043, 92, 49, 99, 101, 304, 105, 104, 55, 111, 63, 73, 89, 116, 61, 120, 125, 78, 129, 188, 67, 93, 137, 97, 18349355, 140, 433

Offset: 0

Alex Ratushnyak, Jun 10 2012

nonn

			Smallest n such that A213381(n)=9 is 17, so a(9)=17.

Cf. A213381, A213861

#include  // GCC -O3 -fopenmp
#include   // 11 minutes
#define SIZE 4096
#define STEP 0x1000000
long long first[SIZE];
int main(int argc, char **argv)
{
    unsigned long long a, i;
    for (a=0; a0; p=(p*p)%b, t>>=1) {
            if (t&1) r=(r*p)%b;
        }
        if (r

a(n) = smallest k>n such that (-2)^k == n (mod k+2).

Terms a(10) onward from Max Alekseyev, Jan 31 2014

A213382 Numbers n such that n^n mod (n + 2) = n.

Original entry on oeis.org

1, 4, 7, 13, 16, 19, 31, 37, 49, 55, 61, 67, 85, 91, 109, 121, 127, 139, 157, 175, 181, 193, 196, 199, 211, 217, 235, 247, 265, 289, 301, 307, 313, 319, 325, 337, 379, 391, 397, 409, 415, 445, 451, 469, 487, 499, 517, 535, 541, 571, 577, 589, 595, 631, 667, 679

Offset: 1

Alex Ratushnyak, Jun 10 2012

nonn

Equivalently, numbers n such that (n^n+2)/(n+2) is an integer. Derek Orr, May 23 2014
It was conjectured that A176003 is a subsequence.
Terms that do not appear in A176003: 16, 61, 193, 196, 313, 397, 691, 729, 769 ...
The conjecture is correct: verify the cases 1 and 3, then it suffices to show that (3p-2)^(3p-2) = 3p-2 mod 3 and mod p. Mod 3 the congruence is 1^(3p-2) = 1, and mod p the congruence is (-2)^(3p-2) = -2 which is true by Fermat's little theorem. - Charles R Greathouse IV, Sep 12 2012
a(62) = 729 is the first number not congruent to 1 mod 3. - Derek Orr, May 23 2014

			A213381(n) = 7^7 mod 9 = 7, so 7 is in the sequence.

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

Cf. A213381 : a(n) = n^n mod (n+2).

Cf. A176003.

Select[Range[700],PowerMod[#,#,#+2]==#&] (* Harvey P. Dale, Oct 03 2015 *)

is(n)=Mod(n,n+2)^n==n \\ Charles R Greathouse IV, Sep 12 2012

for n in range(999):
    x = n**n % (n+2)
    if x==n:
        print(n, end=", ")

A212844 a(n) = 2^(n+2) mod n.

Original entry on oeis.org

0, 0, 2, 0, 3, 4, 1, 0, 5, 6, 8, 4, 8, 2, 2, 0, 8, 4, 8, 4, 11, 16, 8, 16, 3, 16, 23, 8, 8, 16, 8, 0, 32, 16, 2, 4, 8, 16, 32, 24, 8, 4, 8, 20, 23, 16, 8, 16, 22, 46, 32, 12, 8, 4, 7, 16, 32, 16, 8, 4, 8, 16, 32, 0, 63, 58, 8, 64, 32, 36, 8, 40, 8, 16, 47

Offset: 1

Alex Ratushnyak, Jul 22 2012

nonn

Also a(n) = x^x mod (x-2), where x = n+2.
Indices of 0's: 2^k, k>=0.
Indices of 1's: 7, 511, 713, 11023, 15553, 43873, 81079, 95263, 323593, 628153, 2275183, 6520633, 6955513, 7947583, 10817233, 12627943, 14223823, 15346303, 19852423, 27923663, 28529473, ...
Conjecture: every integer k >= 0 appears in a(n) at least once.
Each number below 69 appears at least once. Some large first occurrences: a(39806401) = 25, a(259274569) = 33, a(10571927) = 55, a(18039353) = 81. - Charles R Greathouse IV, Jul 21 2015

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

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

Cf. A000312, A015910, A062173, A112983, A213381, A213859.

A212844 := proc(n)
    modp( 2&^ (n+2),n) ;
end proc: # R. J. Mathar, Jul 24 2012

Table[PowerMod[2, n+2, n], {n, 79}] (* Alonso del Arte, Jul 22 2012 *)

A212844(n)=lift(Mod(2,n)^(n+2)) \\ M. F. Hasler, Jul 23 2012

for n in range(1,99):
    print(2**(n+2) % n, end=',')

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

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=',')

A213085 First occurrence of n in A212844.

Original entry on oeis.org

1, 7, 3, 5, 6, 9, 10, 55, 11, 3521, 246, 21, 52, 89969, 286, 60827, 22, 57481, 1501, 31937, 44, 2977, 49, 27, 40, 39806401, 110, 16777, 114, 214293, 24823, 247, 33, 259274569, 222, 2739, 70, 5993, 253217, 1062899, 72, 2007, 215, 85, 140, 4187, 50, 75

Offset: 0

Alex Ratushnyak, Jul 22 2012

nonn

It is conjectured that every integer n>=0 appears in A212844 at least once, and therefore every a(n) is defined.
Indices of terms that are bigger than 2^32-1 and possibly undefined: 69, 91, 114, 127, 141, 157, 175, 181, 195, 301, 313, 339, ...
Indices 69, 127, 175, 181, 301, 313, 339, ... correspond to terms that either do not exist or are greater than 2*10^12. - Charles R Greathouse IV, Aug 13 2015

			Smallest n such that A212844(n)=1 is 7, so a(1)=7.

Charles R Greathouse IV, Table of n, a(n) for n = 0..68

Cf. A212844, A213381, A213407.

a(n) = my(k=1); while(lift(Mod(2, k)^(k+2)) != n, k++); k; \\ Michel Marcus, Aug 14 2015

cp's OEIS Frontend

A213407 First occurrence of n in A213381.

Views

Author

Keywords

Examples

Crossrefs

Programs

C

Formula

Extensions

A213382 Numbers n such that n^n mod (n + 2) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A212844 a(n) = 2^(n+2) mod n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A213085 First occurrence of n in A212844.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI