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A029790 None of the digits in n is present in n^2 or n^3. 2
2, 3, 7, 8, 22, 47, 53, 77, 92, 157, 187, 188, 192, 552, 558, 577, 707, 772, 922, 2522, 8338, 17177, 66888, 575757, 929522, 1717177, 8888588 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If it exists, a(28) > 10^9. - Derek Orr, Jul 13 2014

a(28) > 2 * 10^9. - Pratik Koirala, Eugene Fiorini, Otis Tweneboah, Nathan Fox, Jul 13 2015

From Manfred Scheucher, Jul 23 2015 (Start)

I strongly conjecture that a(28) does not exist. Assume that a(28)  exists, then a(28) > 10^20 must hold. Moreover, it must have the form as stated in the output of my script. "valid prefix" states which digits may occur:

???0...00002 valid prefix: [0, 1, 2, 3, 5, 6, 7, 9]

???0...00092 valid prefix: [0, 1, 2, 3, 5, 9]

???0...00003 valid prefix: [0, 1, 3, 4, 5, 6, 8]

???0...00007 valid prefix: [0, 1, 2, 5, 6, 7, 8]

???0...00707 valid prefix: [0, 1, 6, 7]

???0...00157 valid prefix: [0, 1, 5, 7]

???0...00077 valid prefix: [0, 1, 7, 8]

???0...00008 valid prefix: [0, 3, 7, 8, 9]

???0...00558 valid prefix: [0, 5, 8, 9]

???0...00188 valid prefix: [0, 1, 8, 9]

At this point I think that it can be easily shown by hand that none of those "subnumbers" yield a(28). F.e., assume that a(28)=x=???100...0002, then x^2=???200..004. Another example: x=???300...008 admits x^2=???800..0064.

(End)

LINKS

Table of n, a(n) for n=1..27.

_Manfred Scheucher_, Python Script

EXAMPLE

For n = 47, n^2 = 2209 and n^3 = 103823. 4 and 7 do not appear in either of these numbers.

MAPLE

filter:= proc(n) local S1, S23;

S1:= convert(convert(n, base, 10), set);

S23:= convert(convert(n^2, base, 10), set) union

      convert(convert(n^3, base, 10), set);

nops(S1 intersect S23)=0

end proc:

select(filter, [$1..10^5]); # Robert Israel, Jul 14 2014

MATHEMATICA

Select[Range@ 1000000, Intersection[IntegerDigits[#^2], IntegerDigits@ #] == {} && Intersection[IntegerDigits[#^3], IntegerDigits@ #] == {} &] (* Michael De Vlieger, Jul 23 2015 *)

PROG

(Python)

def a(n):

..s = str(n)

..s2 = str(n**2)

..s3 = str(n**3)

..count = 0

..for i in s:

....if s2.count(i) == 0 and s3.count(i) == 0:

......count += 1

....else:

......break

..if count == len(s):

....return True

n = 1

while n < 10**9:

..if a(n):

....print(n, end=', ')

..n += 1

# Derek Orr, Jul 13 2014

(PARI) isok(n) = d = vecsort(Set(digits(n))); dd = vecsort(Set(digits(n^2))); ddd = vecsort(Set(digits(n^3))); for (i=1, #d, if (vecsearch(dd, d[i]) || vecsearch(ddd, d[i]), return (0)); ); 1 \\ Michel Marcus, Jul 13 2014

(PARI) is(n)=#setintersect(setunion(Set(digits(n^2)), Set(digits(n^3))), Set(digits(n)))==0 \\ Charles R Greathouse IV, Jul 23 2015

CROSSREFS

Sequence in context: A024540 A029785 A045545 * A206725 A129645 A112994

Adjacent sequences:  A029787 A029788 A029789 * A029791 A029792 A029793

KEYWORD

nonn,base,more,hard

AUTHOR

Patrick De Geest

STATUS

approved

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Last modified January 19 20:33 EST 2019. Contains 319310 sequences. (Running on oeis4.)