A360570 Numbers m such that m concatenated with k produces a cube for some 0 <= k <= m.

6, 12, 21, 34, 49, 51, 58, 68, 72, 92, 100, 106, 121, 133, 138, 156, 172, 175, 195, 196, 219, 243, 262, 274, 297, 327, 337, 359, 373, 405, 409, 428, 466, 491, 506, 531, 548, 551, 583, 593, 614, 658, 681, 685, 689, 740, 753, 778, 800, 804, 830, 851, 857, 884

Offset: 1

Author

Chai Wah Wu, Feb 20 2023

Keywords

• nonn
• base

Comments

Numbers k such that A243092(k) <> A245631(k).

Examples

			64 = 4^3 is a cube and 4 <= 6, thus 6 is a term.
343 = 7^3 is a cube and 3 <= 34, thus 34 is a term.
1000 = 10^3 is a cube and 0 <= 100, thus 100 is a term.
5359375 = 175^3 is a cube and 375 <= 5359, thus 5359 is a term.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A243092, A245631.

Programs

Maple

filter:= proc(n) local d,a,b,r,rmin,i;
  for d from 1 to ilog10(n)+1 do
    a:=ceil((10^d*n)^(1/3));
    b:= floor((10^d*(n+1))^(1/3));
 if d = 1 then rmin:= 0 else rmin:= 10^(d-1) fi;
    for i from a to b do
      r:= i^3 -10^d*n;
      if r >= min(10^d,n+1) then break fi;
      if r < rmin then next fi;
      return true
    od
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 22 2023

PARI

is(n) = { for (k=0, n, if (ispower(n*10^max(1,#digits(k))+k, 3), return (1))); return (0) } \\ Rémy Sigrist, Feb 20 2023

Python

from itertools import count, islice
from sympy import integer_nthroot
def A360570_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        if integer_nthroot(m:=10*n,3)[1]:
            yield n
        else:
            a = 1
            while (k:=(integer_nthroot(a*(m+1)-1,3)[0]+1)**3-m*a)>=10*a:
                a *= 10
                if a > n:
                    break
            else:
                if k <= n:
                    yield n
A360570_list = list(islice(A360570_gen(),20))