A081765 -id:A081765 - cp's OEIS frontend

A251603 Numbers k such that k + 2 divides k^k - 2.

3, 4551, 46775, 82503, 106976, 1642796, 4290771, 4492203, 4976427, 21537831, 21549347, 21879936, 51127259, 56786087, 60296571, 80837771, 87761787, 94424463, 96593696, 138644871, 168864999, 221395539, 255881451, 297460451, 305198247, 360306363, 562654203

Offset: 1

Juri-Stepan Gerasimov, Dec 05 2014

nonn

Numbers k such that (k^k - 2)/(k + 2) is an integer.
Since k == -2 (mod k+2), also numbers k such that k + 2 divides (-2)^k - 2. - Robert Israel, Jan 04 2015
Numbers k == 0 (mod 4) such that A066602(k/2+1) = 8, and odd numbers k such that k = 3 or A082493(k+2) = 8. - Robert Israel, Apr 08 2015

			3 is in this sequence because 3 + 2 = 5 divides 3^3 - 2 = 25.

Max Alekseyev, Table of n, a(n) for n = 1..890 (all terms below 10^15)

Cf. A001477, A004273, A004275, A066602, A082493, A081765, A213382, A252606, A357125.

[n: n in [0..10000] | Denominator((n^n-2)/(n+2)) eq 1];

isA251603 := proc(n)
    if modp(n &^ n-2,n+2) = 0 then
        true;
    else
        false;
    end if;
end proc:
A251603 := proc(n)
    option remember;
    local a;
    if n = 1 then
        3;
    else
        for a from procname(n-1)+1 do
            if isA251603(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jan 09 2015

Select[Range[10^6], Mod[PowerMod[#, #, # + 2] - 2, # + 2] == 0 &] (* Michael De Vlieger, Dec 20 2014, based on Robert G. Wilson v at A252041 *)

for(n=1,10^9,if(Mod(n,n+2)^n==+2,print1(n,", "))); \\ Joerg Arndt, Dec 06 2014

A251603_list = [n for n in range(1,10**6) if pow(n, n, n+2) == 2] # Chai Wah Wu, Apr 13 2015

The even terms form A122711, the odd terms are those in A245319 (forming A357125) decreased by 2. - Max Alekseyev, Sep 22 2016

a(6)-a(27) from Joerg Arndt, Dec 06 2014

A122711 Even numbers n such that n+2 divides n+2^n.

Original entry on oeis.org

106976, 1642796, 21879936, 96593696, 6926872352, 21235295216, 24936246176, 25867010016, 80832867116, 82230049056, 208329074876, 360598467776, 533800559216, 587627376176, 661575990912, 662312961696, 664490433776, 737374205276, 831623487276, 1052816473676, 1137732817376, 1213045642656, 1270015920636

Offset: 1

Zak Seidov, Sep 23 2006

nonn

Same as even numbers n such that 2^n == 2 (mod n+2). - Robert G. Wilson v, Sep 27 2006
n must be a multiple of 4. A002326(n/4) must not be divisible by 2 or 3. If p is an odd prime factor of n+2, (n+2)/p mod A002326((p-1)/2)=3. - Martin Fuller, Oct 09 2006
Also, the positive numbers A015922(k)-2 that are multiples of 4. E.g., a(1) = 106976 = A015922(3926)-2. Hence, a(n)+2 forms a subsequence of A015922 (and of A130134) consisting of the terms congruent to 2 modulo 4. - Max Alekseyev, Apr 03 2014

Max Alekseyev, Table of n, a(n) for n = 1..110 (all terms below 10^15)

Cf. A081765, A015922, A122042.

Do[ If[ PowerMod[2, 2n, 2n + 2] == 2, Print@2n], {n, 10^9}] (* Robert G. Wilson v, Sep 27 2006 *)

More terms from Max Alekseyev, Sep 23 2006, Oct 01 2006
More terms from Martin Fuller, Oct 09 2006
Terms a(18) onward from Max Alekseyev, Apr 09 2014
b-file corrected by Max Alekseyev, Oct 11 2016

A252606 Numbers j such that j + 2 divides 2^j + 2.

Original entry on oeis.org

3, 4, 16, 196, 2836, 4551, 5956, 25936, 46775, 65536, 82503, 540736, 598816, 797476, 1151536, 3704416, 4290771, 4492203, 4976427, 8095984, 11272276, 13362420, 21235696, 21537831, 21549347, 29640832, 31084096, 42913396, 49960912, 51127259, 55137316, 56786087, 60296571, 70254724, 70836676

Offset: 1

Juri-Stepan Gerasimov, Mar 03 2015

Numbers j such that (2^j + 2)/(j + 2) is an integer. Numbers j such that (2^j - j)/(j + 2) is an integer.
From Robert Israel, Apr 09 2015: (Start)
The even members of this sequence (4, 16, 196, 2836, ...) are the numbers 2*k-2 where k>=3 is odd and 4^k == -8 (mod k).
The odd members of this sequence (3, 4551, 46775, 82503, ...) are the numbers k-2 where k>=3 is odd and 2^k == -8 (mod k). (End)
2^m is in this sequence for m = (2, 4, 16, 36, 120, 256, 456, 1296, 2556, ...), with the subsequence m = 2^k, k = (1, 2, 4, 8, 16, ...). - M. F. Hasler, Apr 09 2015

			3 is in this sequence because (2^3 + 2)/(3 + 2) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..78

Cf. A001477, A004273, A004275, A081765, A213382, A251603.

[n: n in [0..1200000] | Denominator((2^n+2)/(n+2)) eq 1];

select(t -> 2 &^t + 2 mod (t + 2) = 0, [$1..10^6]); # Robert Israel, Apr 09 2015

Select[Range[10^6],IntegerQ[(2^#+2)/(#+2)]&] (* Ivan N. Ianakiev, Apr 17 2015 *)

for(n=1,10^5,if((2^n+2)%(n+2)==0,print1(n,", "))) \\ Derek Orr, Apr 05 2015

is(n)=Mod(2,n+2)^n==-2 \\ M. F. Hasler, Apr 09 2015

A252606_list = [n for n in range(10**4) if pow(2, n, n+2) == n] # Chai Wah Wu, Apr 16 2015

a(17)-a(22) from Tom Edgar, Mar 03 2015

More terms from Chai Wah Wu, Apr 16 2015

A122042 Numbers n such that (2^n+n)/(n+2) is prime.

Original entry on oeis.org

13, 19, 55, 469, 3385

Offset: 1

Zak Seidov, Sep 14 2006

All terms are also in A081765 = numbers n such that n+2 divides 2^(n-1)-1.

Cf. A081765.

Do[If[PrimeQ[(2^n+n)/(n+2)],Print[n]],{n,1,5000}]

is(n)=ispseudoprime((2^n+n)/(n+2)) \\ Charles R Greathouse IV, Jun 13 2017

a(5) from Alexander Adamchuk, Sep 17 2006

A271267 Even numbers k such that k + 2 divides k^k + 2.

Original entry on oeis.org

4, 16, 196, 2836, 5956, 25936, 65536, 540736, 598816, 797476, 1151536, 3704416, 8095984, 11272276, 13362420, 21235696, 29640832, 31084096, 42913396, 49960912, 55137316, 70254724, 70836676, 81158416, 94618996, 111849956, 129275056, 150026176, 168267856, 169242676, 189796420, 192226516, 198464176, 208232116, 244553296, 246605776, 300018016, 318143296

Offset: 1

Altug Alkan, Apr 03 2016

nonn

In other words, even numbers k such that k + 2 divides A014566(k) + 1.
Even terms of A213382.
4, 16, 65536 are the numbers of the form 2^(2^(2^k)), for k >= 0. Are there other members of this sequence with the form of 2^(2^(2^k))?
2^(2^(2^3)) and 2^(2^(2^4)) are terms. - Michael S. Branicky, Apr 16 2021

			4 is a term because 4 + 2 = 6 divides 4^4 + 2 = 258.

Michael S. Branicky, Table of n, a(n) for n = 1..150

Cf. A014566, A081765, A213382.

Select[Range[2, 10^4, 2], Divisible[#^# + 2, # + 2] &] (* Michael De Vlieger, Apr 03 2016 *)

lista(nn) = forstep(n=2, nn, 2, if( Mod(n, n+2)^n == -2 , print1(n, ", "))); \\ Joerg Arndt, Apr 03 2016

def afind(limit):
  k = 2
  while k < limit:
    if (pow(k, k, k+2) + 2)%(k+2) == 0: print(k, end=", ")
    k += 2
afind(10**7) # Michael S. Branicky, Apr 16 2021

cp's OEIS Frontend

A251603 Numbers k such that k + 2 divides k^k - 2.

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A122711 Even numbers n such that n+2 divides n+2^n.

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A252606 Numbers j such that j + 2 divides 2^j + 2.

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A122042 Numbers n such that (2^n+n)/(n+2) is prime.

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A271267 Even numbers k such that k + 2 divides k^k + 2.

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Programs

Mathematica

PARI

Python