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
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
Programs
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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))
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