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A061341 Numbers not ending in 0 whose cubes are concatenations of other cubes. 2
22, 303, 1006, 2272, 3003, 6001, 6006, 10006, 30003, 50015, 50024, 60001, 60006, 60025, 100006, 120015, 121925, 150005, 150012, 240005, 300003, 466085, 500015, 500024, 600001, 600006, 600025, 600075, 1000006, 1200015, 1500005, 1500012, 1500015, 2400005, 2500006 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From François Marques, Jul 11 2021: (Start)

If a and b are integers such that 3*a^2*b and 3*a*b^2 can be obtained as concatenations of cubes then a*10^k+b is a term in the list for k greater than the maximum of the number of digits in b^3, 3*a^2*b and 3*a*b^2.

Conjecture: the last digit of a(n) is in {1, 2, 3, 4, 5, 6, 8} and if it's 3 then there is k>=2 such that a(n) = 3*10^k+3.

(End)

LINKS

François Marques, Table of n, a(n) for n = 1..100

EXAMPLE

2272^3 = 11728027648 = 1_1728_0_27_64_8 (where the underscores indicate concatenation).

PROG

(PARI) can_split(v, b=10, p=3)=if(#v==0, 1, for(k=1, #v, if(ispower(fromdigits(v[1..k], b), p) && can_split(v[k+1..#v], b, p), return(1))); return(0));

isok(n, b=10, p=3)=if(n%b==0, return(0)); my(v=digits(n^p, b)); for(k=1, #v-1, if(ispower(fromdigits(v[1..k], b), p) && can_split(v[k+1..#v], b, p), return(1))); return(0); \\ François Marques, Jul 12 2021

(Python)

from sympy import integer_nthroot

def iscube(n): return integer_nthroot(n, 3)[1]

def ok3(n, c):

    if n%10 == 9 or (c == 1 and n%10 == 0): return False

    if c > 1 and iscube(n): return True

    d = str(n)

    for i in range(1, len(d)):

        if iscube(int(d[:i])) and ok3(int(d[i:]), c+1): return True

    return False

print([r for r in range(61000) if ok3(r**3, 1)]) # Michael S. Branicky, Jul 11 2021

CROSSREFS

Cf. A000578, A009421.

Sequence in context: A155785 A077525 A083765 * A088279 A225898 A028109

Adjacent sequences:  A061338 A061339 A061340 * A061342 A061343 A061344

KEYWORD

base,nonn

AUTHOR

Erich Friedman, Jun 06 2001

STATUS

approved

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)