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 A248970 Bases that lack a three-digit narcissistic number: numbers n with no 1 <= x < n, 0 <= y,z < n such that x^3 + y^3 + z^3 = n^2*x + n*y + z. 1

%I

%S 72,90,108,153,270,423,450,531,558,630,648,738,1044,1098,1125,1224,

%T 1242,1287,1440,1503,1566,1611,1620,1800,1935,2034,2142,2250,2358,

%U 2439,2448,2511,2754,2790,2799,2862,2943,2952

%N Bases that lack a three-digit narcissistic number: numbers n with no 1 <= x < n, 0 <= y,z < n such that x^3 + y^3 + z^3 = n^2*x + n*y + z.

%C All terms are multiples of 9.

%H Charles R Greathouse IV, <a href="/A248970/b248970.txt">Table of n, a(n) for n = 1..296</a>

%H StackExchange, <a href="http://math.stackexchange.com/questions/425901/narcissistic-numbers-in-other-bases">Proof that all terms are multiples of 9</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NarcissisticNumber.html">Narcissistic Number</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Narcissistic_number">Narcissistic number</a>

%e 17=1^3+2^3+2^3 is 122 in base 3, so 3 is not in the sequence.

%o (MATLAB) for Base=1:100,Noneyet=1;for a=1:Base-1,for b=0:Base-1,for c=0:Base-1,if a*Base*Base+b*Base+c==a^3+b^3+c^3,Noneyet = 0;end;end;end;end;if Noneyet,disp(Base);end;end;

%o (PARI) is(n)=if(n%9,return(0)); for(x=1,n-1,for(y=0,x,for(z=0,y, my(v=digits(x^3+y^3+z^3,n)); if(vecsort(v)==[z,y,x],return(0))))); 1 \\ slow; _Charles R Greathouse IV_, Oct 21 2014

%o (PARI) is(n)=if(n%9,return(0)); my(mx=n*(n-1)*(n-2),t); for(x=1,n-1,for(y=0,n-1, t=n*(n*x+y)-x^3-y^3; if(t>=0 && t <= mx && !polisirreducible('z^3-'z-t) && #select(P->poldegree(P)==1&&polcoeff(P,0)<=0 && polcoeff(P,0)>-n, factor('z^3-'z-t)[,1]), return(0)))); 1 \\ faster; _Charles R Greathouse IV_, Oct 21 2014

%Y Cf. A005188 (base 10).

%K nonn,base

%O 1,1

%A _Michael R Peake_, Oct 18 2014

%E a(26)-a(38) from _Charles R Greathouse IV_, Oct 21 2014

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)