A056790 -id:A056790 - cp's OEIS frontend

A007571 a(n) = largest prime factor of n^n + 1.

2, 5, 7, 257, 521, 97, 911, 673, 530713, 27961, 58367, 2227777, 79301, 176597, 142111, 67280421310721, 45957792327018709121, 33388093, 870542161121, 4406613081041681, 22864311556633, 73194743542229, 1522029233, 27250359649

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Tyler Busby, Table of n, a(n) for n = 1..148 (terms 1..123 from Hugo Pfoertner, terms 124..135 from Yurii Ivanov)
factordb, Status of 148^148+1 ... 167^167+1.

Cf. A006530, A006486, A014566, A056790.

[Max(PrimeDivisors(n^n+1)):n in [1..24]]; // Marius A. Burtea, Aug 24 2019

Table[ FactorInteger[ n^n + 1, FactorComplete -> True ] [ [ -1, 1 ] ], {n, 1, 25} ]

for(k=1, 24, my(x=factor(k^k+1), f=x[#x[, 1], 1]); print1(f,", ")) \\ Hugo Pfoertner, Aug 23 2019

a(n) = A006530(A014566(n)). - Michel Marcus, Aug 24 2019

A056788 a(n) = n^n + (n-1)^(n-1).

Original entry on oeis.org

2, 5, 31, 283, 3381, 49781, 870199, 17600759, 404197705, 10387420489, 295311670611, 9201412118867, 311791207040509, 11414881932150269, 449005897206417391, 18884637964090410991, 845687005960046315793, 40173648337182874339601, 2017766063735610126699403

Offset: 1

Walter Nissen, Aug 20 2000

nonn

For even n > 1, the absolute value of the discriminant of the polynomial x^n+x-1. [Corrected by Artur Jasinski, May 07 2010]

The largest known prime in this sequence is a(4) = 283.

			a(3) = 2^2 + 3^3 = 4 + 27 = 31.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

Pierre Letouzey, Generalized Hofstadter functions G, H and beyond: numeration systems and discrepancy, arXiv:2502.12615 [cs.DM], 2025. See p. 18.
Walter Nissen, post on np ( n ) = n^n + (n+1)^(n+1), on home page "Up for the count!". (Updated Oct 02 2012)

Cf. A000312 (n^n), A086797 (discriminant of the polynomial x^n-x-1).

Cf. A056187, A056790, A192397 (smallest & largest prime factor of a(n), records of the latter), A217435 = bigomega(a(n)).

Join[{2}, Table[n^n+(n-1)^(n-1), {n, 2, 20}]] (* T. D. Noe, Aug 13 2004 *)
Join[{2},Total/@Partition[Table[n^n,{n,20}],2,1]] (* Harvey P. Dale, Jun 26 2017 *)

A056788(n)=n^n+(n-1)^(n-1)  \\ M. F. Hasler, Oct 02 2012

Minor corrections by M. F. Hasler, Oct 02 2012

A056187 Least prime factor of n^n + (n+1)^(n+1).

Original entry on oeis.org

2, 5, 31, 283, 3, 67, 11, 11, 5, 173, 3, 1237, 7, 31, 7334881, 227, 3, 773, 149, 47, 11, 5, 3, 101, 13, 73, 151, 349, 3, 4421, 107, 191, 17, 7, 3, 17, 19, 624808693, 2273, 25788481, 3, 5, 59, 1752761753, 23, 2144707, 3, 49547, 5, 275851515609829434269, 19, 919, 3, 13, 7, 107, 29

Offset: 0

Walter Nissen, Aug 20 2000

nonn

			a(5) = 67 because 5^5 + 6^6 = 3125 + 46656 = 67 * 743.

Hugo Pfoertner, Table of n, a(n) for n = 0..234, (first 96 terms from Chai Wah Wu).
factordb.com, Status of 235^235+236^236.
Walter Nissen, np(n) = n^n + (n+1)^(n+1) -- 2 prominent questions. (Updated Oct 02 2012)

Cf. A056788, A056790, A192397, A217435.

A056187(n)=factor((n+1)^(n+1)+n^n)[1,1] \\ M. F. Hasler, Oct 04 2012

a(n) = A020639(A056788(n+1)). - M. F. Hasler, Oct 04 2012

2 added by Arkadiusz Wesolowski, Jun 30 2011

a(53)-a(56) from Chai Wah Wu, Jul 22 2019

A192397 Record holders for greatest prime factor of n^n + (n+1)^(n+1).

Original entry on oeis.org

2, 5, 31, 283, 743, 1600069, 60042893, 7438489991, 61215157711, 34041259347101651, 6564253087266573169, 22022174223585405703, 69454092876521107983605569601, 2360926164108571968813424783598971267, 462605180698333957063188362720170172617217, 14645575916792712592989131451003587034531413111, 214236369415820799335832514547376967536187180963

Offset: 1

Walter Nissen, Jun 29 2011

nonn

			60042893 = A056790(9) is in the sequence because all earlier members of A056790 are smaller than 60042893.

Walter Nissen, More terms with additional commentary

Cf. A056790, A056187.

fmax=0;for(k=0,35,my(x=factor(k^k+(k+1)^(k+1)),f=x[#x[,1],1]);if(f>fmax,print1(f,", ");fmax=f)) \\ Hugo Pfoertner, Aug 18 2019

A056790(m) < A056790(n), for all m < n

2 added by Arkadiusz Wesolowski, Jun 30 2011

A309747 Numbers k such that k^k + (k+1)^(k+1) is a semiprime.

Original entry on oeis.org

5, 7, 9, 11, 14, 21, 37, 38, 39, 57, 90, 97, 162

Offset: 1

Hugo Pfoertner, Aug 15 2019

Numbers k such that A217435(k+1) = 2.

a(14) >= 235, see FactorDB link.

			a(1) = 5 because 5^5 + 6^6 = 49781 = 67*743.

FactorDB, Status of 235^235+236^236.
Walter Nissen, Addendum to Prime factors of n^n + (n+1)^(n+1), Table F, 2014-01-07.

Cf. A056187, A056788, A056790, A217435.

for(k=0,40,if(bigomega(k^k+(k+1)^(k+1))==2,print1(k,", ")))

cp's OEIS Frontend

A007571 a(n) = largest prime factor of n^n + 1.

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A056788 a(n) = n^n + (n-1)^(n-1).

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A056187 Least prime factor of n^n + (n+1)^(n+1).

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A192397 Record holders for greatest prime factor of n^n + (n+1)^(n+1).

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A309747 Numbers k such that k^k + (k+1)^(k+1) is a semiprime.

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PARI