A076116 - OEIS

%I #20 Jul 09 2024 08:41:57

%S 1,13,8,0,23,2,46,0,20,8,116,0,163,18,218,6,281,32,352,0,431,50,518,0,

%T 28,72,14,0,827,98,946,0,1073,128,1208,0,1351,162,1502,0,1661,200,

%U 1828,0,53,242,2186,98,32,43,2576,0,2783,36,2998,0,3221,392,3452,0,3691,450

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

%H Robert Israel, <a href="/A076116/b076116.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Robert Israel_, Nov 15 2023: (Start)

%F 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.

%F 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)

%p f:= proc(n) local y,F,t,k,v;

%p       if n::odd then

%p          F:= ifactors(n)[2];

%p          y:= mul(t[1]^ceil(t[2]/3),t=F);

%p          k:= 1+floor((n*(n-1)/2)^(1/3)/y);

%p          (k*y)^3/n - (n-1)/2;

%p       else

%p          v:= padic:-ordp(n,2);

%p          if v mod 3 <> 1 then return 0 fi;

%p          F:= ifactors(n/2^v)[2];

%p          y:= mul(t[1]^ceil(t[2]/3),t=F)*2^((v-1)/3);

%p          k:= 1 + floor((n*(n-1)/2)^(1/3)/y);

%p          if k::even then k:= k+1 fi;

%p          (k*y)^3/n - (n-1)/2;

%p       fi

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Nov 15 2023

%t f[n_] := Module[{y, F, t, k, v},

%t If[OddQ[n],

%t    F = FactorInteger[n];

%t    y = Product[t[[1]]^Ceiling[t[[2]]/3], {t, F}];

%t    k = 1 + Floor[(n*(n-1)/2)^(1/3)/y];

%t    (k*y)^3/n - (n-1)/2

%t ,

%t    v = IntegerExponent[n, 2];

%t    If[Mod[v, 3] != 1, Return[0]];

%t    F = FactorInteger[n/2^v];

%t    y = Product[t[[1]]^Ceiling[t[[2]]/3], {t, F}]*2^((v-1)/3);

%t    k = 1 + Floor[(n*(n-1)/2)^(1/3)/y];

%t    If[EvenQ[k], k = k+1];

%t    (k*y)^3/n - (n-1)/2]];

%t Map[f, Range[100]] (* _Jean-François Alcover_, Jul 09 2024, after _Robert Israel_ *)

%Y Cf. A007814, A019555, A076117, A076114.

%K nonn,look

%O 1,2

%A _Amarnath Murthy_, Oct 09 2002

%E More terms from _David Wasserman_, Apr 02 2005