A056788 -id:A056788 - cp's OEIS frontend

A097396 Duplicate of A056788.

Original entry on oeis.org

2, 5, 31, 283, 3381, 49781, 870199, 17600759, 404197705, 10387420489

Offset: 1

dead

A162928 Primes of the form A056788(k)-2.

Original entry on oeis.org

3, 29, 281, 870197, 311791207040507

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 17 2009

nonn

The term a(6) has 291 digits.

			a(1)=A056788(2)-2=3. a(2)=A056788(3)-2=29, associated with k=2, 3, 4, 7, 13, 136,....

Cf. A162927, A056788

f[n_]:=n^n+(n+1)^(n+1)-2; lst={};Do[If[PrimeQ[f[n]],AppendTo[lst,f[n]]], {n,2*5!}];lst

Edited by R. J. Mathar, Jul 19 2009

A056790 Greatest prime factor of n^n + (n+1)^(n+1).

Original entry on oeis.org

2, 5, 31, 283, 23, 743, 331, 1600069, 410353, 60042893, 8969, 7438489991, 116803, 4879633159, 61215157711, 338142271, 34041259347101651, 45072130459, 6564253087266573169, 22022174223585405703, 121937899012999, 69454092876521107983605569601, 5311242856728321929909

Offset: 0

Walter Nissen, Aug 20 2000

Note that n^n + (n+1)^(n+1) = A056788(n+1).
Becomes "hard" (unknown) around n ~ 112, cf. link: As of today, even A217435(113) (number of prime factors) is unknown. - M. F. Hasler, Oct 04 2012
As of today, the first unknown term is a(143). - Daniel Suteu, Mar 11 2019

			a(4) = 23 because 4^4 + 5^5 = 3381 = 3 * 7^2 * 23.

Daniel Suteu, Table of n, a(n) for n = 0..142
Walter Nissen, np(n) = n^n + (n+1)^(n+1) -- 2 prominent questions. (Updated Oct 02 2012)

Cf. A056187, A192397.

Cf. A217435.

Join[{2},FactorInteger[Total[#]][[-1,1]]&/@Partition[Table[n^n,{n,30}],2,1]] (* Harvey P. Dale, Apr 21 2018 *)

A056790(n)=vecmax(factor((n+1)^(n+1)+n^n)[,1])  \\ M. F. Hasler, Oct 04 2012

a(n) = A006530(A056788(n+1)). - M. F. Hasler, Oct 04 2012

a(0) = 2 added by Arkadiusz Wesolowski, Jun 30 2011

a(21)-a(22) added by Daniel Suteu, Mar 11 2019

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

A217435 Number of prime factors of n^n+(n-1)^(n-1), counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 3, 2, 3, 2, 5, 2, 6, 4, 2, 3, 6, 3, 3, 4, 6, 2, 7, 3, 5, 5, 3, 3, 5, 3, 3, 3, 7, 4, 5, 4, 3, 2, 2, 2, 6, 3, 6, 4, 5, 3, 8, 6, 6, 3, 5, 5, 5, 5, 3, 3, 5, 2, 8, 4, 4, 6, 6, 6, 8, 8, 4, 3, 4, 7, 7, 6, 6, 4, 4, 7, 9, 6, 7, 5, 9, 5, 7, 4, 3, 5, 6, 7, 9, 3, 2, 5

Offset: 1

M. F. Hasler, Oct 02 2012

M. F. Hasler and Amiram Eldar, Table of n, a(n) for n = 1..143 (terms 1..112 from M. F. Hasler)
W. Nissen, np ( n ) = n^n + (n+1)^(n+1) — 2 prominent questions, Oct 02 2012.
W. Nissen, Prime factors of np ( n ) = n^n + (n+1)^(n+1), Up for the Count.

A217435:=n->numtheory[bigomega](n^n+(n-1)^(n-1)): seq(A217435(n), n=1..30); # Wesley Ivan Hurt, Jan 27 2017

Join[{1}, Table[PrimeOmega[n^n + (n-1)^(n-1)], {n, 2, 30}]] (* Amiram Eldar, Feb 24 2020 *)
Join[{1},PrimeOmega[Total/@Partition[#^#&/@Range[30],2,1]]] (* Harvey P. Dale, Oct 02 2021 *)

for(n=1,999,print1(bigomega((n-1)^(n-1)+n^n)","))

a(n) = A001222(A056788(n)).

Data beyond a(49) from W. Nissen's web site.

A086797 Discriminant of the polynomial x^n - x - 1.

Original entry on oeis.org

0, 5, -23, -283, 2869, 49781, -776887, -17600759, 370643273, 10387420489, -275311670611, -9201412118867, 293959006143997, 11414881932150269, -426781883555301359, -18884637964090410991, 808793517812627212561, 40173648337182874339601

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

sign

Selmer proved that for all n the Galois group of the polynomial x^n - x - 1 over the rationals is the symmetric group S_n. [Comment corrected by Artur Jasinski, Feb 06 2007]

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
H. Osada, The Galois groups of the polynomials X^n + a X^l + b, Journal of Number Theory, Feb. 1987, vol. 25, (no.2):230-8.
Ernst S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.

Cf. A000312 (n^n), A007781 (n^n - (n-1)^(n-1)), A056788 (n^n + (n-1)^(n-1)).

Cf. A086783.

PARI
```
a(n)=poldisc(x^n-x-1)
```

Except for the sign, the sequence alternates between the sum and difference of consecutive terms of A000312. a(n) = (n^n + (-1)^n (n-1)^(n-1))*(-1)^ceiling(1+n/2). - T. D. Noe, Aug 13 2004

More terms from Benoit Cloitre, Aug 06 2003

A114654 Discriminant of the polynomial x^n + x + 1.

Original entry on oeis.org

1, -3, -31, 229, 3381, -43531, -870199, 15953673, 404197705, -9612579511, -295311670611, 8630788777645, 311791207040509, -10809131718965763, -449005897206417391, 18008850183328692241, 845687005960046315793, -38519167813410200811247

Offset: 1

T. D. Noe, Dec 21 2005

sign

Except for the sign, the sequence alternates between the sum and difference of consecutive terms of A000312. x^2+x+1 divides x^n+x+1 for n=2 (mod 3).

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Cf. A000312 (n^n), A007781 (n^n - (n-1)^(n-1)), A056788 (n^n + (n-1)^(n-1)), A086797 (discriminant of the polynomial x^n-x-1).

Table[Discriminant[x^n + x + 1, x], {n, 0, 100}] (* Artur Jasinski, Oct 12 2007 *)

a(n) = poldisc(x^n+x+1); \\ Michel Marcus, Aug 28 2020

for n>1, a(n) = (n^n + (-1)^(n-1) * (n-1)^(n-1)) * (-1)^floor(n/2).

a(n) = (Cos[Pi n/2]+Sin[Pi n/2])(n^n)+(Cos[Pi(n+1)/2]+Sin[Pi(n+1)/2])(n+1)^(n+1). - Artur Jasinski, Oct 12 2007

A162929 Primes of the form n^n+(n+1)^(n+1)+6.

Original entry on oeis.org

11, 37, 49787, 404197711

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 17 2009

nonn

Primes of the form A056788(k)+6, associated with k=2, 3, 6, 9, 126,...

The term a(5) has 265 digits.

			2^2+3^3+6=37.

Cf. A162927, A056788

f[n_]:=n^n+(n+1)^(n+1)+6; lst={};Do[If[PrimeQ[f[n]],AppendTo[lst,f[n]]], {n,2*5!}];lst
Select[Table[n^n+(n+1)^(n+1)+6,{n,500}],PrimeQ] (* Harvey P. Dale, Jan 30 2014 *)

Edited by R. J. Mathar, Jul 19 2009

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

A097396 Duplicate of A056788.

Views

Author

Keywords

A162928 Primes of the form A056788(k)-2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A056790 Greatest prime factor of n^n + (n+1)^(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A217435 Number of prime factors of n^n+(n-1)^(n-1), counted with multiplicity.

Views

Author

Keywords

Links

Programs

Maple

Mathematica

PARI

Formula

Extensions

A086797 Discriminant of the polynomial x^n - x - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A114654 Discriminant of the polynomial x^n + x + 1.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

PARI

Formula

A162929 Primes of the form n^n+(n+1)^(n+1)+6.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs