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A007377

Numbers k such that the decimal expansion of 2^k contains no 0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, 31, 32, 33, 34, 35, 36, 37, 39, 49, 51, 67, 72, 76, 77, 81, 86

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

It is an open problem of long standing to show that 86 is the last term.

A027870(a(n)) = A224782(a(n)) = 0. - Reinhard Zumkeller, Apr 30 2013

See A030700 for the analog for 3^k, which seems to end with k=68. - M. F. Hasler, Mar 07 2014

Checked up to k = 10^10. - David Radcliffe, Aug 21 2022

REFERENCES

J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..36.

W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]

Eric Weisstein's World of Mathematics, Zero

R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993

EXAMPLE

Here is 2^86, conjecturally the largest power of 2 not containing a 0: 77371252455336267181195264. - N. J. A. Sloane, Feb 10 2023

MAPLE

remove(t -> has(convert(2^t, base, 10), 0), [$0..1000]); # Robert Israel, Dec 29 2015

MATHEMATICA

Do[ If[ Union[ RealDigits[ 2^n ] [[1]]] [[1]] != 0, Print[ n ] ], {n, 1, 60000}]

Select[Range@1000, First@Union@IntegerDigits[2^# ] != 0 &]

Select[Range[0, 100], DigitCount[2^#, 10, 0]==0&] (* Harvey P. Dale, Feb 06 2015 *)

PROG

(Magma) [ n: n in [0..50000] | not 0 in Intseq(2^n) ]; // Bruno Berselli, Jun 08 2011

(Perl) use bignum;

for(0..99) {

if((1<<$_) =~ /^[1-9]+$/) {

print "$_, "

}

} # Charles R Greathouse IV, Jun 30 2011

(PARI) for(n=0, 99, if(vecmin(eval(Vec(Str(2^n)))), print1(n", "))) \\ Charles R Greathouse IV, Jun 30 2011

(Haskell)

import Data.List (elemIndices)

a007377 n = a007377_list !! (n-1)

a007377_list = elemIndices 0 a027870_list

-- Reinhard Zumkeller, Apr 30 2013

(Python)

def ok(n): return '0' not in str(2**n)

print(list(filter(ok, range(10**4)))) # Michael S. Branicky, Aug 08 2021

CROSSREFS

Cf. A027870, A030700, A102483, A034293, A224782.

Some similar sequences are listed in A035064.

Cf. also A031142.

Sequence in context: A174887 A092598 A247811 * A305932 A213882 A135140

Adjacent sequences: A007374 A007375 A007376 * A007378 A007379 A007380

KEYWORD

nonn,base,more

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

a(1) = 0 prepended by Reinhard Zumkeller, Apr 30 2013

STATUS

approved

URL: https://oeis.org/A007377

⇱ A007377 - OEIS