

A255869


Least m > 0 such that gcd(m^n+19, (m+1)^n+19) > 1, or 0 if there is no such m.


20



1, 0, 3, 2408, 1, 3976, 608, 28, 1, 88, 23, 464658, 1, 319924724, 3, 7, 1, 1628, 138, 2219409, 1, 6, 5, 594, 1, 872, 3, 92, 1, 392, 65, 2278155, 1, 3755866, 4793, 13, 1, 7873, 3, 614294, 1, 448812437, 5
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OFFSET

0,3


COMMENTS

See A118119, which is the main entry for this class of sequences.
a(43) <= 8153777984244162781089834.  Max Alekseyev, Aug 06 2015


LINKS

Table of n, a(n) for n=0..42.


FORMULA

a(4k) = 1 for k>=0, because gcd(1^(4k)+19, 2^(4k)+19) = gcd(20, 16^k1) >= 5 since 16 = 1 (mod 5).


EXAMPLE

For n=0 and n=4, see formula with k=0 resp. k=1.
For n=1, gcd(m^n+19, (m+1)^n+19) = gcd(m+19, m+20) = 1, therefore a(1)=0.
For n=2, gcd(3^2+19, 4^2+19) = 7 and (m,m+1) = (3,4) is the smallest pair which yields a GCD > 1 here.


MATHEMATICA

A255869[n_] := Module[{m = 1}, While[GCD[m^n + 19, (m + 1)^n + 19] <= 1, m++]; m]; Join[{1, 0}, Table[A255869[n], {n, 2, 12}]] (* Robert Price, Oct 16 2018 *)


PROG

(PARI) a(n, c=19, L=10^7, S=1)={n!=1 && for(a=S, L, gcd(a^n+c, (a+1)^n+c)>1 && return(a))}


CROSSREFS

Cf. A118119, A255832, A255852A255868
Sequence in context: A293099 A286715 A081176 * A289650 A171361 A203687
Adjacent sequences: A255866 A255867 A255868 * A255870 A255871 A255872


KEYWORD

nonn,hard


AUTHOR

M. F. Hasler, Mar 09 2015


EXTENSIONS

a(13)a(40) from Hiroaki Yamanouchi, Mar 12 2015
a(41)a(42) from Max Alekseyev, Aug 06 2015


STATUS

approved



