A242787 -id:A242787 - cp's OEIS frontend

A242788 Numbers n such that (n^n-3)/(n-3) is an integer.

1, 2, 4, 5, 6, 7, 9, 11, 13, 15, 16, 27, 31, 33, 36, 55, 73, 91, 133, 241, 249, 366, 367, 491, 513, 577, 733, 757, 871, 913, 971, 991, 1233, 1333, 1576, 1711, 1927, 2071, 2346, 2593, 2731, 3307, 3391, 3529, 4005, 4591, 5113, 5371, 5409, 5671, 5793, 6567, 6801, 7465, 7591

Offset: 1

Derek Orr, May 22 2014

nonn

For n > 6, equivalent to n such that n^n = 3 mod n-3. - Chai Wah Wu, Jan 19 2015

			(6^6-3)/(6-3) = 46653/3 = 15551 is an integer. Thus 6 is a member of this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..4388

Cf. A242787.

Select[ Range@ 7600, Mod[ PowerMod[#, #, # - 3] - 3, # - 3] == 0 &] (* Robert G. Wilson v, Jan 21 2015 *)

for(n=1,10^4,if(n!=3,s=(n^n-3)/(n-3);if(floor(s)==s,print(n))))

A242788_list = [1,2,4,5,6] + [n for n in range(7,10**6) if pow(n, n, n-3) == 3]
# Chai Wah Wu, Jan 19 2015

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

A242789 Least number k > 1 such that (k^k-n)/(k-n) is an integer.

Original entry on oeis.org

2, 3, 2, 2, 3, 3, 3, 4, 3, 4, 3, 6, 4, 7, 3, 4, 5, 4, 7, 5, 5, 4, 7, 9, 4, 7, 3, 7, 5, 13, 5, 4, 9, 12, 5, 6, 10, 16, 9, 4, 9, 16, 7, 5, 5, 4, 10, 13, 7, 7, 11, 13, 5, 9, 7, 6, 5, 12, 19, 9, 11, 17, 7, 7, 5, 11, 4, 16, 9, 5, 11, 16, 9, 13, 15, 13, 9, 12, 7, 31, 6, 16, 5

Offset: 1

Derek Orr, May 22 2014

nonn

a(n) <= n+1 for all n.

			(2^2-8)/(2-8) = -4/-6 is not an integer. (3^3-8)/(3-8) = 19/-5 is not an integer. (4^4-8)/(4-8) = 248/4 = 62 is an integer. Thus a(8) = 4.

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

Cf. A242787, A242788.

a[n_] := Module[{k = 2}, While[k == n || ! Divisible[k^k - n, k - n], k++]; k]; Array[a, 100] (* Amiram Eldar, Jun 04 2021 *)

a(n)=for(k=2,n+1,if(k!=n,s=(k^k-n)/(k-n);if(floor(s)==s,return(k))));
n=1;while(n<100,print(a(n));n+=1)

A242796 Least number k such that (k^k-n)/(k-n) is prime, or 0 if no such k exists.

Original entry on oeis.org

2, 4, 6, 5, 0, 13, 9, 11, 0, 12, 22, 13, 37, 28, 36, 0, 0, 171, 73, 85, 0, 0, 0, 29, 0, 0, 0, 0, 517, 35, 40, 49, 44, 49, 0, 41, 46, 40, 0, 0, 51, 0, 52, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2841, 0, 0, 0, 0, 0, 0, 67, 0, 64, 0, 199, 125, 221, 0, 0, 153, 113, 239, 0, 97, 0, 0, 0

Offset: 1

Derek Orr, May 22 2014

a(n)=0 is confirmed for k <= 5000.

			(1^1-2)/(1-2) = 1 is not prime. (3^3-2)/(3-2) = 25 is not prime. (4^4-2)/(4-2) = 254/2 = 127 is prime. Thus a(2) = 4.

Cf. A242787, A242788.

a(n)=for(k=1,5000,if(k!=n,s=(k^k-n)/(k-n);if(floor(s)==s,if(ispseudoprime(s),return(k)))));
n=1;while(n<100,print(a(n));n+=1)

We don't normally allow conjectural terms, except in special circumstances. This is one of those exceptions, for if we included only terms that are known for certain, not much of this sequence would remain. - N. J. A. Sloane, May 31 2014

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A242788 Numbers n such that (n^n-3)/(n-3) is an integer.

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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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A242789 Least number k > 1 such that (k^k-n)/(k-n) is an integer.

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A242796 Least number k such that (k^k-n)/(k-n) is prime, or 0 if no such k exists.

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