A076989 Smallest cube of the form n*k + 1 with k>0.

8, 27, 64, 125, 216, 343, 8, 729, 64, 1331, 1728, 2197, 27, 729, 4096, 4913, 5832, 343, 343, 9261, 64, 12167, 13824, 15625, 17576, 27, 1000, 729, 27000, 29791, 125, 35937, 39304, 42875, 1331, 2197, 1000, 343, 4096, 68921, 74088, 15625, 216, 91125, 4096

Offset: 1

Author

Amarnath Murthy, Oct 25 2002

Keywords

• nonn
• easy
• look

Examples

			a(9) = 64 as 64 = 7*9 + 1.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A066498, A076947.

Programs

Maple

a[1] := 8:for n from 2 to 150 do j := 2:while((j^3 mod n)<>1)do j := j+1:od: a[n] := j^3:od:seq(a[k],k=1..150);
# Alternative
f:=proc(n) local R;
  R:= sort([numtheory:-rootsunity(3,n)] mod n);
  if nops(R)=1 then (n+1)^3 else R[2]^3 fi
end proc:
map(f, [$1..150]); # Robert Israel, Mar 30 2018

Mathematica

sc[n_]:=Module[{k=1},While[!IntegerQ[Surd[n*k+1,3]],k++];n*k+1]; Array[ sc,50] (* Harvey P. Dale, Mar 30 2018 *)

PARI

first(n) = my(res = vector(n)); {res[1] = 8; for(i = 2, n + 1, i3 = i ^ 3-1; d = divisors(i3); j = 2; while(j <= #d && d[j] <= n, if(res[d[j]] == 0, res[d[j]] = i3 + 1); j++)); res} \\ David A. Corneth, Mar 30 2018

Formula

a(n) <= (n+1)^3. In particular, a(n) < (n+1)^3 if n is in A066498. - David A. Corneth, Mar 30 2018

a(n) = A076947(n)*n + 1. - Altug Alkan, Mar 30 2018

Extensions

More terms from Sascha Kurz, Jan 26 2003