A116892 -id:A116892 - cp's OEIS frontend

A116893 Numbers k such that gcd(k!+1, k^k+1) > 1.

1, 3, 23, 39, 51, 63, 95, 99, 131, 183, 191, 215, 239, 251, 299, 303, 315, 363, 371, 411, 419, 431, 443, 495, 543, 575, 659, 683, 711, 743, 755, 791, 831, 891, 911, 935, 975, 1019, 1031, 1055, 1071, 1143, 1155, 1191, 1211, 1223, 1251, 1275, 1295, 1355

Offset: 1

Giovanni Resta, Mar 01 2006

See A116892 for the corresponding values of the GCD. See also comments in A116891.

			gcd(1!+1, 1^1+1) = 2, gcd(2!+1, 2^2+1) = 1 and gcd(3!+1, 3^3+1) = 7, so 1 and 3 are the first two terms of the sequence.

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

Cf. A014566, A038507, A067658, A116891, A116892, A116894.

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See Links section.
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Select[Range[1500], (GCD[ #!+1, #^#+1] > 1)&]

isok(n) = gcd(n! + 1, n^n + 1) != 1; \\ Michel Marcus, Jul 22 2018

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

Original entry on oeis.org

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

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

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A116893 Numbers k such that gcd(k!+1, k^k+1) > 1.

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