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.
72, 90, 108, 153, 270, 423, 450, 531, 558, 630, 648, 738, 1044, 1098, 1125, 1224, 1242, 1287, 1440, 1503, 1566, 1611, 1620, 1800, 1935, 2034, 2142, 2250, 2358, 2439, 2448, 2511, 2754, 2790, 2799, 2862, 2943, 2952
Offset: 1
Examples
17=1^3+2^3+2^3 is 122 in base 3, so 3 is not in the sequence.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..296
- StackExchange, Proof that all terms are multiples of 9
- Eric Weisstein's World of Mathematics, Narcissistic Number
- Wikipedia, Narcissistic number
Crossrefs
Cf. A005188 (base 10).
Programs
-
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;
-
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
-
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
Extensions
a(26)-a(38) from Charles R Greathouse IV, Oct 21 2014
Comments