A076116 Start of the smallest string of n consecutive positive numbers with a cube sum, or 0 if no such number exists.

1, 13, 8, 0, 23, 2, 46, 0, 20, 8, 116, 0, 163, 18, 218, 6, 281, 32, 352, 0, 431, 50, 518, 0, 28, 72, 14, 0, 827, 98, 946, 0, 1073, 128, 1208, 0, 1351, 162, 1502, 0, 1661, 200, 1828, 0, 53, 242, 2186, 98, 32, 43, 2576, 0, 2783, 36, 2998, 0, 3221, 392, 3452, 0, 3691, 450

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Robert Israel, Nov 15 2023: (Start)

If n is odd, then a(n) is the least positive integer of the form (k*A019555(n))^3/n - (n-1)/2 where k is an integer.

If n is even, then let v = A007814(n). If v == 1 (mod 3) then a(n) is the least positive integer of the form (k*A019555(n/2))^3/n - (n-1)/2 where k an odd integer; otherwise, a(n) = 0. (End)

Maple

f:= proc(n) local y, F, t, k, v;
      if n::odd then
         F:= ifactors(n)[2];
         y:= mul(t[1]^ceil(t[2]/3), t=F);
         k:= 1+floor((n*(n-1)/2)^(1/3)/y);
         (k*y)^3/n - (n-1)/2;
      else
         v:= padic:-ordp(n, 2);
         if v mod 3 <> 1 then return 0 fi;
         F:= ifactors(n/2^v)[2];
         y:= mul(t[1]^ceil(t[2]/3), t=F)*2^((v-1)/3);
         k:= 1 + floor((n*(n-1)/2)^(1/3)/y);
         if k::even then k:= k+1 fi;
         (k*y)^3/n - (n-1)/2;
      fi
end proc:
map(f, [$1..100]); # Robert Israel, Nov 15 2023

Mathematica

f[n_] := Module[{y, F, t, k, v},
If[OddQ[n],
   F = FactorInteger[n];
   y = Product[t[[1]]^Ceiling[t[[2]]/3], {t, F}];
   k = 1 + Floor[(n*(n-1)/2)^(1/3)/y];
   (k*y)^3/n - (n-1)/2
,
   v = IntegerExponent[n, 2];
   If[Mod[v, 3] != 1, Return[0]];
   F = FactorInteger[n/2^v];
   y = Product[t[[1]]^Ceiling[t[[2]]/3], {t, F}]*2^((v-1)/3);
   k = 1 + Floor[(n*(n-1)/2)^(1/3)/y];
   If[EvenQ[k], k = k+1];
   (k*y)^3/n - (n-1)/2]];
Map[f, Range[100]] (* Jean-François Alcover, Jul 09 2024, after Robert Israel *)

Crossrefs

Cf. A007814, A019555, A076117, A076114.

Sequence in context: A110056 A159562 A249024 * A010216 A302456 A303238

Adjacent sequences: A076113 A076114 A076115 * A076117 A076118 A076119

Keyword

nonn,look

Author

Amarnath Murthy, Oct 09 2002

Extensions

More terms from David Wasserman, Apr 02 2005

Status

approved