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A251731 Least k such that k^3 + q is divisible by 3^n where q is the n-th number congruent to 1 or -1 (mod 18).

%I A251731 #25 Feb 14 2020 03:22:09
%S A251731 2,1,2,16,32,145,62,1363,3458,19492,58928,89308,70028,1594318,1890551,
%T A251731 189871,31401806,47918575,190704887,163454602,502048577,9481323661,
%U A251731 11627845304,34656488290,115450061084,286130228125,2303721331049,1569269836240,22013516320412
%N A251731 Least k such that k^3 + q is divisible by 3^n where q is the n-th number congruent to 1 or -1 (mod 18).
%C A251731 It is known that k always exists if q is congruent to +-1 mod 18.
%C A251731 The numbers congruent to 1 or -1 (mod 18) are 1, 17, 19, 35, 37, ... = {A161705} UNION {A239129}.
%C A251731 For n >= 2, k^3 == (9 - 18*n - 7*(-1)^n)/2 (mod 3^n) if and only if k - a(n) is divisible by 3^(n-1). - _Jinyuan Wang_, Feb 13 2020
%H A251731 Robert Israel, <a href="/A251731/b251731.txt">Table of n, a(n) for n = 1..2095</a>
%e A251731 a(1) = 2 because the first number of the form +-1 (mod 18) is 1, and 2^3 + 1 = 9 = 3*3^1;
%e A251731 a(2) = 1 because the second number of the form +-1 (mod 18) is 17, and 1^3 + 17 = 18 = 2*3^2;
%e A251731 a(3) = 2 because the third number of the form +-1 (mod 18) is 19, and 2^3 + 19 = 27 = 3^3;
%e A251731 a(4)= 16 because the fourth number of the form +-1 (mod 18) is 35, and 16^3 + 35 = 4131 = 51*3^4.
%p A251731 f:= proc(n) local q,R,k;
%p A251731   if n::odd then q:= 9*n-8 else q:= 9*n-1 fi;
%p A251731   min(map(subs,[msolve(k^3+q,3^n)],k))
%p A251731 end proc:
%p A251731 map(f, [$1..30]); # _Robert Israel_, Dec 23 2018
%t A251731 lst1={1};Do[lst1=Union[lst1,Union[{18*n+1},{18*n-1}]],{n,1,10}];lst={};Do[k=1;While[Mod[k^3+lst1[[n]],3^n]!=0,k++];Print[n," ",k],{n,1,10}];lst
%o A251731 (PARI) a(n) = {if (n % 2, q = 9*(n-1)+1, q = 9*n-1); m = 3^n; k = 1; while ((k^3+q) % m, k++); k;} \\ _Michel Marcus_, Jan 07 2015
%Y A251731 Cf. A161705, A239129.
%K A251731 nonn
%O A251731 1,1
%A A251731 _Michel Lagneau_, Dec 07 2014