A213382 -id:A213382 - cp's OEIS frontend

A242801 Least number k > 1 such that (k^k+n)/(k+n) is an integer.

3, 4, 3, 6, 3, 8, 5, 4, 3, 4, 5, 7, 11, 4, 5, 18, 4, 20, 5, 8, 3, 11, 9, 4, 5, 13, 9, 16, 7, 19, 7, 4, 11, 5, 5, 7, 19, 4, 9, 16, 7, 9, 5, 6, 15, 16, 5, 8, 7, 7, 9, 13, 19, 12, 5, 7, 12, 29, 4, 5, 16, 16, 9, 10, 7, 16, 13, 16, 6, 17, 9, 13, 5, 16, 5, 9, 7, 13, 7, 4, 9, 41, 15

Offset: 1

Derek Orr, May 23 2014

nonn

It is believed that a(n) <= n+2 for all n > 0.

a(n) also exists for all n < 1. - Robert G. Wilson v, Jun 05 2014

			(2^2+1)/(2+1) = 5/3 is not an integer. (3^3+1)/(3+1) = 28/4 = 7 is an integer. Thus a(1) = 3.

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

Cf. A213382, A242800.

f[n_] := Block[{k = 2}, While[ Mod[ PowerMod[k, k, k + n] + n, k + n] != 0, k++]; k]; Array[f, 90] (* Robert G. Wilson v, Jun 05 2014 *)

a(n)=for(k=2,1000,s=(k^k+n)/(k+n);if(floor(s)==s,return(k)));
n=1;while(n<100,print(a(n), ", ");n+=1) \\ corrected by Michel Marcus, May 24 2014

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

Original entry on oeis.org

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

A252041 Numbers m such that m - 3 divides m^m + 3.

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 85, 105, 136, 186, 262, 820, 1161, 2626, 2926, 4924, 10396, 11656, 19689, 27637, 33736, 36046, 42886, 42901, 53866, 55189, 82741, 95266, 103762, 106822, 127401, 135460, 251506, 366796, 375220, 413326, 466966, 531445, 553456, 568876

Offset: 1

Juri-Stepan Gerasimov, Dec 12 2014

nonn

Numbers m such that (m^m + 3)/(m - 3) is an integer.
Most but not all terms are congruent to 4 modulo 6. - Robert G. Wilson v, Dec 19 2014
Note that m^m == 3^m (mod m-3). - Robert Israel, Dec 19 2014

			2 is in this sequence because (2^2 + 3)/(2 - 3) = -7 is an integer.
4 is in this sequence because (4^4 + 3)/(4 - 3) = 259 is an integer.
7 is not in the sequence because (7^7 + 3)/4 = 411773/2, which is not an integer.

Robert G. Wilson v, Table of n, a(n) for n = 1..331

Cf. ...............Numbers n such that x divides y, where:
...x......y....k = 0.....k = 1.....k = 2......k = 3.......
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751..this sequence..
..n+k..n^n-k..A000027...A004275...A251603....A251862......
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Dec 28 2014)

[n: n in [4..50000] | Denominator((n^n+3)/(n-3)) eq 1];

select(t -> 3 &^t + 3 mod (t-3) = 0, [1,2,$4..10^6]); # Robert Israel, Dec 19 2014

fQ[n_] := Mod[PowerMod[n, n, n - 3] + 3, n - 3] == 0; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Dec 13 2014; modified by Robert G. Wilson v, Dec 19 2014 *)

isok(n) = (n != 3) && (Mod(n, n-3)^n  == -3); \\ Michel Marcus, Dec 13 2014

More terms from Michel Marcus, Dec 13 2014

A249751 Numbers m such that m - 2 divides m^m + 2.

Original entry on oeis.org

3, 4, 7, 8, 67, 260, 379, 1191, 1471, 5076, 25807, 58591, 103780, 134947, 137347, 170587, 203236, 272611, 285391, 420211, 453748, 538735, 540856, 592411, 618451, 680707, 778807, 1163067, 1306936, 1520443, 1700947, 1891336, 2099203, 2831011, 3481960, 4020031

Offset: 1

Juri-Stepan Gerasimov, Dec 05 2014

nonn

			3 is in this sequence because (3^3 + 2)/(3 - 2) = 29 is an integer.

Chai Wah Wu, Table of n, a(n) for n = 1..160

Cf. A081762, A242787, A213382, A252041.

[n: n in [3..10000] | Denominator((n^n+2)/(n-2)) eq 1];

fQ[n_] := Mod[ PowerMod[ n, n, n - 2] + 2, n - 2] == 0; Select[ Range@ 4100000, fQ] (* Robert G. Wilson v, Dec 19 2014 *)

A249751_list = [n for n in range(3,10**7) if n==3 or pow(n,n,n-2) == n-4]
# Chai Wah Wu, Dec 06 2014

More terms from Chai Wah Wu, Dec 06 2014

A251862 Numbers m such that m + 3 divides m^m - 3.

Original entry on oeis.org

3, 7, 10, 27, 727, 1587, 9838, 758206, 789223, 1018846, 1588126, 1595287, 2387206, 4263586, 9494746, 12697378, 17379860, 21480726, 25439767, 38541526, 44219926, 55561536, 62072326, 64335356, 70032586, 83142466, 85409276

Offset: 1

Juri-Stepan Gerasimov, Dec 10 2014

nonn

m such that m+3 divides (-3)^m - 3. - Robert Israel, Dec 14 2014

			3 is in this sequence because 3 + 3 = 6 divides 3^3 - 3 = 24.

Chai Wah Wu, Table of n, a(n) for n = 1..96

Cf. ...............Numbers n such that x divides y, where:
...x.....y......k=0.......k=1.......k=2........k=3........
..n-k..n^n-k..A000027...A087156...A242787....A242788......
..n-k..n^n+k..A000027..see below..A249751....A252041......
..n+k..n^n-k..A000027...A004275...A251603..this sequence..
..n+k..n^n+k..A000027...A004273...A213382....A242800......
(For x=n-1 and y=n^n+1, the only terms are 0, 2 and 3. - David L. Harden, Jan 14 2015)

[n: n in [2..10000] | Denominator((n^n-3)/(n+3)) eq 1];

select(t ->((-3) &^ (t) - 3) mod (t+3) = 0, [$1..10^6]); # Robert Israel, Dec 14 2014

a251862[n_] := Select[Range[n], Mod[PowerMod[#, #, # + 3] - 3, # + 3] == 0 &]; a251862[10^6] (* Michael De Vlieger, Dec 14 2014, after Robert G. Wilson v at A252041 *)

isok(n) = Mod(n, n+3)^n == 3; \\ Michel Marcus, Dec 10 2014

A251862_list = [n for n in range(10**6) if pow(-3, n, n+3) == 3] # Chai Wah Wu, Jan 19 2015

[n for n in range(10^4) if (n + 3).divides((-3)^n - 3)] # Peter Luschny, Jan 17 2015

More terms from Michel Marcus, Dec 10 2014

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

A257002 Primes p such that p+2 divides p^p+2.

Original entry on oeis.org

7, 13, 19, 31, 37, 61, 67, 109, 127, 139, 157, 181, 193, 199, 211, 307, 313, 337, 379, 397, 409, 487, 499, 541, 571, 577, 631, 691, 751, 769, 787, 811, 829, 877, 919, 937, 991, 1009, 1021, 1039, 1117, 1201, 1291, 1297, 1327, 1381, 1399, 1459, 1471, 1531, 1567

Offset: 1

K. D. Bajpai, Apr 14 2015

nonn

All the terms in this sequence are congruent to 1 mod 3.

Primes p such that 2^p == 2 (mod p+2). Includes A091180. - Robert Israel, Apr 14 2015

			a(1) = 7 is prime; 7+2 = 9 divides 7^7 + 2 = 823545.
a(2) = 13 is prime; 13+2 = 15 divides 13^13 + 2 = 302875106592255.

K. D. Bajpai, Table of n, a(n) for n = 1..5700

Cf. A000040, A091180, A213382.

[ p: p in PrimesUpTo(1600) | (p^p+2) mod (p+2) eq 0 ]; // Vincenzo Librandi, Apr 15 2015

select(t -> isprime(t) and (2 &^t - 2) mod (t+2) = 0, [seq(6*i+1,i=1..10^4)]); # Robert Israel, Apr 14 2015

Select[Prime[Range[3000]], Mod[#^# + 2, # + 2] == 0 &]
Select[Prime[Range[500]],PowerMod[#,#,#+2]==#&] (* Harvey P. Dale, May 19 2017 *)

forprime(p=2,1000, if(Mod(p^p+2,p+2)==0, print1(p, ", ")));

from sympy import prime
A257002_list = [p for p in (prime(n) for n in range(1,10**4)) if pow(p, p, p+2) == p] # Chai Wah Wu, Apr 14 2015

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

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A242801 Least number k > 1 such that (k^k+n)/(k+n) is an integer.

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

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A252041 Numbers m such that m - 3 divides m^m + 3.

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A249751 Numbers m such that m - 2 divides m^m + 2.

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A251862 Numbers m such that m + 3 divides m^m - 3.

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

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