A116893 -id:A116893 - cp's OEIS frontend

A116891 a(n) = gcd(n! + 1, n^n + 1).

2, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 103, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 191, 1, 1, 1, 199, 1, 1

Offset: 1

Giovanni Resta, Mar 01 2006

Apparently all the values greater than 1 (cf. A116892) are prime numbers and are equal to 2n+1 with only 4 exceptions for n<82000 (cf. A116894).
From Antti Karttunen, Jul 22 2018: (Start)
The first duplicated value > 1 is 157519 = a(43755) = a(78759). Note that 43755 = 15*2917, while 78759 = 27*2917.
It seems that for the long time after a(1) = 2, all other terms > 1 occur only at such positions k that k+1 is not squarefree. However, this turns out to be false as a(208161) = 555097, and 208162 is a squarefree number.
(End)

			a(3) = gcd(3! + 1, 3^3 + 1) = gcd(7,28) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..80001

Cf. A014566, A038507, A067658, A116892, A116893, A116894.

Table[GCD[n! + 1, n^n + 1], {n, 101}] (* Robert G. Wilson v, Mar 09 2006 *)

A116891(n) = gcd(n!+1,(n^n)+1); \\ Antti Karttunen, Jul 22 2018

A116892 Values of gcd(k!+1, k^k+1), when greater than 1.

Original entry on oeis.org

2, 7, 47, 79, 103, 127, 191, 199, 263, 367, 383, 431, 479, 503, 599, 607, 631, 727, 743, 823, 839, 863, 887, 991, 1087, 1151, 1319, 1367, 1423, 1487, 1511, 1583, 1663, 1783, 1823, 1871, 1951, 2039, 2063, 2111, 2143, 2287, 2311, 2383, 2423, 2447, 2503, 2551

Offset: 1

Giovanni Resta, Mar 01 2006

Apart from the initial term (2) and few exceptional values (A116894) this sequence seems to coincide with A067658. The values of k for which the terms of this sequence are obtained are in A116893.

			gcd(1!+1,1^1+1) = 2 gives the first term;
gcd(3!+1,3^3+1) = gcd(7,28) = 7 gives the second, and so on.

Nick Hobson, Table of n, a(n) for n = 1..10000 (first 1832 terms from Antti Karttunen)

Cf. A014566, A038507, A067658, A116891, A116893, A116894.

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See Links section in A116893.
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f[n_] := GCD[n! + 1, n^n + 1]; t = Array[f, 1295]; Rest@ Union@ t (* Robert G. Wilson v, Mar 09 2006 *)

lista(nn) = for (n=1, nn, if ((g=gcd(n! + 1, n^n + 1)) != 1, print1(g, ", "))); \\ Michel Marcus, Jul 22 2018

Entries checked by Robert G. Wilson v, Mar 09 2006

A116894 Numbers k such that gcd(k! + 1, k^k + 1) is neither 1 nor 2k+1.

Original entry on oeis.org

1, 5427, 41255, 43755, 208161, 496175, 497135

Offset: 1

Giovanni Resta, Mar 01 2006

g(n) = gcd(n! + 1, n^n + 1) is almost always equal to 1 or to 2n+1. These are the known exceptions: g(1) = 2, g(5427) = 10453, g(41255) = 129341, g(43755) = 157519, g(208161) = 555097. - Hans Havermann, Mar 28 2006

a(8) > 1000000. - Nick Hobson, Feb 20 2024

			gcd(1! + 1, 1^1 + 1) = 2 and 2 != 2*1 + 1, so 1 belongs to the sequence.

Cf. A014566, A038507, A067658, A116891, A116892, A116893.

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// See Links section in A116893.
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a(5) from Hans Havermann, Mar 28 2006

a(6)-a(7) from Nick Hobson, Feb 20 2024

A134656 Corresponding GCD values in A116894.

Original entry on oeis.org

2, 10453, 129341, 157519, 555097, 595411, 624363411431

Offset: 1

Ryan Propper, Feb 01 2008

Cf. A116894.

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// See Links section in A116893.
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for(n = 1, 10^9, my(x = gcd(n! + 1, n^n + 1)); if(x != 1 && x != 2*n + 1, print1(x, ", ")))

a(n) = gcd(A116894(n)! + 1, A116894(n)^A116894(n) + 1)

a(6)-a(7) from Nick Hobson, Feb 20 2024

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A116891 a(n) = gcd(n! + 1, n^n + 1).

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A116892 Values of gcd(k!+1, k^k+1), when greater than 1.

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A116894 Numbers k such that gcd(k! + 1, k^k + 1) is neither 1 nor 2k+1.

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A134656 Corresponding GCD values in A116894.

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