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A255853 Least k > 0 such that gcd(k^n+3, (k+1)^n+3) > 1, or 0 if there is no such k. 4
1, 0, 6, 56, 3, 29, 96, 1159823, 384, 9, 3, 1994117680, 13, 247, 6, 15, 3, 1256, 4, 25211925041, 15, 5785, 3, 93602696971, 24, 11, 6, 182, 3, 4644, 92, 12506, 9, 13, 3, 484, 2, 420, 6, 130, 3, 16032496, 12 (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) <= 291613846670877. - Max Alekseyev, Aug 07 2015

LINKS

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

FORMULA

For k>=0, a(6k+4)=3 because gcd(3^(6k+4)+3, 4^(6k+4)+3) = gcd(9^(3k+2)+3, 16^(3k+2)+3) and 9 = 16 = 2 (mod 7) and 2^(3k+2)+3 = 2^2+3 = 0 (mod 7), so the GCD is a positive multiple of 7.

EXAMPLE

For n=1, gcd(k^n+3, (k+1)^n+3) = gcd(k+3, k+4) = 1, therefore a(1)=0.

For n=2, we have gcd(6^2+3, 7^2+3) = gcd(39, 52) = 13, and the pair (k,k+1)=(6,7) is the smallest which yields a GCD > 1, therefore a(2)=6.

MATHEMATICA

A255853[n_] := Module[{m = 1}, While[GCD[m^n + 3, (m + 1)^n + 3] <= 1, m++]; m]; Join[{1, 0}, Table[A255853[n], {n, 2, 10}]] (* Robert Price, Oct 15 2018 *)

PROG

(PARI) a(n, c=3, 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, A255852-A255869

Sequence in context: A132689 A323876 A249672 * A183594 A140790 A335217

Adjacent sequences:  A255850 A255851 A255852 * A255854 A255855 A255856

KEYWORD

nonn,hard

AUTHOR

M. F. Hasler, Mar 08 2015

EXTENSIONS

a(11)-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 September 22 14:26 EDT 2020. Contains 337291 sequences. (Running on oeis4.)