Zeroless powers

This page aims to summarize results, serve as an index, and point out some directions to further work in the OEIS on the subject of zeroless powers, i.e., sequences related to the problem of finding powers of numbers which do not contain the digit 0 in their decimal expansion, with several variants possible:

• instead of 2^n, 3^n, ... consider p-smooth numbers,
• instead of zeroless, consider other (one or more) missing digit(s), or pandigital numbers,
• instead of zeroless, consider numbers with a given number of digits '0', or some other given digit(s).
• instead of base 10, consider other bases.

Index to sequences

Exponents n such that xⁿ has no digit 0 in base 10

• Cf. A007377 (x=2), A030700 (x=3), A030701 (x=4), A008839 (x=5), A030702 (x=6), A030703 (x=7), A030704 (x=8), A030705 (x=9), A030706 (x=11), A195944 (x=13). (See next section below for the actual powers)
• Intersections: A007496 = A007377 ∩ A008839 = {n | 2^n and 5^n are zeroless}.
• Last & largest members of the above sequences:

A020665, n^a(n) is (almost certainly) the largest power of n not containing a zero.

This is conjectural. Here are these values as a table:

$x$	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50
$n_{\max }$	86	68	43	58	44	35	27	34	0	41	26	14	34	27	19	27	17	44	0	13	22	10	13	29	15	9	16	14	0	16	7	23	5	17	22	16	10	19	0	9	13	10	6	39	7	8	19	5	0

PARI/GP code:
Nmax(x,L=99,m=0)=for(n=1,L,vecmin(digits(x^n))&&m=n);m \\ checked with larger L, but given the results, L=99 is enough

The powers themselves: Powers xⁿ without digit 0

• Cf. A238938 (x=2), A238939 (x=3), A238940 (x=4), A195948 (x=5), A238936 (x=6), A195908 (x=7), ....., A195946 (x=11), A252482 (x=12), A195945 (x=13).
• Unions of the above: zeroless prime powers: A195942 (excluding primes), A195943 (including primes).

Related sequences

• A052382: Numbers without 0 as a digit, a.k.a. zeroless numbers.
• Least zeroless power with given exponent: A103662(n) = A103663(n)^n.
• A027870: number of 0's in 2^n, ... (analog for other x? other bases?

other digits: 1's A065712, 2's A065710, 3's A065714, 4's A065715, 5's A065716, 6's A065717, 7's A065718, 8's A065719, 9's A065744.)

• A224782: longest run of zero digits in 2^n
• A102483: n s.th. 2^n is zeroless in base 3, ... (other x? other bases?)

Powers with given number of digits d

Not only the zeroless powers appear to be finite, but also the powers with a given number of digits 0, or some other digit. For any base b, digit d between 0 and b-1, basis x, we can consider:

N(x^k,d,b) = number of digits d in x^k written in base b
S(n,x,d,b) = { k ≥ 0 | N(x^k,d,b) = n } = exponents k such that x^k has n digits d in base b
m(n,x,d,b) = min S(n,x,d,b) : smallest k such that x^k has exactly n digits d in base b
M(n,x,d,b) = max S(n,x,d,b) : largest k such that x^k has exactly n digits d in base b
L(n,x,d,b) = # S(n,x,d,b) :  number of powers of x having exactly n digits d in base b

For given x, d, b, each exponent k ≥ 0 is in exactly one of the sets S(n,x,d,b), therefore { S(n,x,d,b); n ≥ 0 } \ {{}} is a partition of the nonnegative integers, which is nontrivial except for a few pathological cases (e.g., if x = 0 or x = 10^m).

Sequences A305932, A305933, A305924, ..., A305929 list in row n the exponents k for which x^k has n digits 0, for x = 2, ..., 9.

The row lengths are mostly in the range 5..50.

Read as flattened sequence, these tables are permutations of the nonnegative integers.

Of course, all data is so far "conjectural", but see the Khovanova link to get an idea of the probability of them being correct.

Further investigations

(to be completed)

p-smooth numbers instead of powers

It has been suggested to consider zeroless p-smooth numbers instead of pure powers.

(to be developed...)

To do

• edit / complete the OEIS entries :
  • change title "Decimal expansion of x^n contains no zeros" to "Numbers n such that ..."
  • add PARI code is_Axxx or (rather?) a simple for(... print(...)) or select(...,vector(...)).
  • add missing Ref's / Links (see below)
  • add missing Crossref's
  • correct offset from 0 to 1 and add initial n=0 where missing.

References

• J. S. Madachy, Mathematics on Vacation, Scribner's, NY, 1966, p. 126.
• T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
• C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
• W. Schneider, NoZeros: Powers n^k without Digit Zero, local copy of www.wschnei.de/digit-related-numbers/nozeros.html (on web.archive.org, as of 7.4.2005), "last modified on 30/1/2003".
• Eric Weisstein's World of Mathematics, Zero

Authors & contributors to this page

• Initial version written by M. F. Hasler 14:22, 7 March 2014 (UTC)

Cite this page as

M. F. Hasler, Zeroless powers.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/Zeroless_powers]