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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 (list; graph; refs; listen; history; text; internal format)
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
FORMULA
a(4k) = 1 for k>=0, because gcd(1^(4k)+19, 2^(4k)+19) = gcd(20, 16^k-1) >= 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
Sequence in context: A286715 A365706 A081176 * A289650 A352600 A171361
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

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)