A305938 -id:A305938 - cp's OEIS frontend

A071531 Smallest exponent r such that n^r contains at least one zero digit (in base 10).

10, 10, 5, 8, 9, 4, 4, 5, 1, 5, 4, 6, 7, 4, 3, 7, 4, 4, 1, 5, 3, 6, 6, 4, 6, 5, 5, 4, 1, 6, 2, 2, 3, 4, 5, 3, 4, 5, 1, 5, 3, 3, 4, 2, 5, 2, 2, 2, 1, 2, 2, 2, 4, 2, 5, 4, 6, 3, 1, 5, 6, 3, 2, 4, 6, 3, 9, 3, 1, 2, 6, 3, 3, 4, 8, 4, 2, 3, 1, 4, 5, 5, 2, 4, 3, 3, 6, 3, 1, 5, 5, 3, 3, 2, 7, 2, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 2

Paul Stoeber (paul.stoeber(AT)stud.tu-ilmenau.de), Jun 02 2002

For all n, a(n) is at most 40000, as shown below. Is 10 an upper bound?
If n has d digits, the numbers n, n^2, ..., n^k have a total of about N = k*(k+1)*d/2, and if these were chosen randomly the probability of having no zeros would be (9/10)^N. The expected number of d-digit numbers n with f(n)>k would be 9*10^(d-1)*(9/10)^N. If k >= 7, (9/10)^(k*(k+1)/2)*10 < 1 so we would expect heuristically that there should be only finitely many n with f(n) > 7. - Robert Israel, Jan 15 2015
The similar definition using "...exactly one digit 0..." would be ill-defined for all multiples of 100 and others (1001, ...). - M. F. Hasler, Jun 25 2018
When r=40000, one of the last five digits of n^r is always 0. Working modulo 10^5, we have 2^r=9736 and 5^r=90625, and both of these are idempotent; also, if gcd(n,10)=1, then n^r=1, and if 10|n, then n^r=0. Therefore the last five digits of n^r are always either 00000, 00001, 09736, or 90625. In particular, a(n) <= 40000. - Mikhail Lavrov, Nov 18 2021

			a(4)=5 because 4^1=4, 4^2=16, 4^3=64, 4^4=256, 4^5=1024 (has zero digit).

Robert Israel, Table of n, a(n) for n = 2..10000
OEIS Wiki, Zeroless powers (2014).

Cf. A305941 for the actual powers n^k.
Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944: decimal expansion of k^n contains no zeros, k = 2, 3, 4, ...
Cf. A305932, A305933, A305924, ..., A305929: row n = {k: x^k has n 0's}, x = 2, 3, ..., 9.
Cf. A305942, ..., A305947, A305938, A305939: #{k: x^k has n 0's}, x = 2, 3, ..., 9.
Cf. A306112, ..., A306119: largest k: x^k has n 0's; x = 2, 3, ..., 9.

f:= proc(n) local j;
for j from 1 do if has(convert(n^j,base,10),0) then return j fi od:
end proc:
seq(f(n),n=2..100); # Robert Israel, Jan 15 2015

zd[n_]:=Module[{r=1},While[DigitCount[n^r,10,0]==0,r++];r]; Array[zd,110,2] (* Harvey P. Dale, Apr 15 2012 *)

A071531(n)=for(k=1, oo, vecmin(digits(n^k))||return(k)) \\ M. F. Hasler, Jun 23 2018

def a(n):
    r, p = 1, n
    while 1:
        if "0" in str(p):
            return r
        r += 1
        p *= n
[a(n) for n in range(2, 100)] # Tim Peters, May 19 2005

a(n) >= 1 with equality iff n is in A011540 \ {0} = {10, 20, ..., 100, 101, ...}. - M. F. Hasler, Jun 23 2018

A305939 Number of powers of 9 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

12, 7, 18, 3, 9, 13, 11, 11, 6, 9, 17, 15, 12, 9, 11, 6, 9, 9, 9, 13, 16, 9, 10, 7, 7, 9, 9, 13, 14, 15, 14, 15, 9, 9, 8, 8, 15, 11, 11, 12, 5, 12, 14, 5, 7, 14, 10, 8, 5, 16, 12

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 12 is the number of terms in A030705 and in A195945, which includes the power 7^0 = 1.

These are the row lengths of A305929. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030705 = row 0 of A305929: k such that 9^k has no 0's; A195945: these powers 9^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063626 = column 1 of A305929: least k such that 9^k has n digits 0 in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938 (analog for 8^k).

A305939(n,M=99*n+199,x=9)=sum(k=0,M,#select(d->!d,digits(x^k))==n)

A305939_vec(nMax,M=99*nMax+199,x=9,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]++);a[^-1]}

A305947 Number of powers of 7 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

10, 11, 12, 13, 9, 10, 9, 7, 10, 14, 21, 10, 18, 7, 11, 11, 12, 15, 17, 10, 11, 6, 10, 16, 13, 9, 7, 9, 11, 12, 10, 16, 7, 16, 9, 14, 13, 13, 9, 17, 14, 12, 11, 9, 13, 9, 12, 12, 9, 12, 14

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 10 is the number of terms in A030703 and in A195908, which includes the power 7^0 = 1.

These are the row lengths of A305927. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030703 (= row 0 of A305927): k such that 7^k has no 0's; A195908: these powers 7^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063606 (= column 1 of A305927): least k such that 7^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305946, A305938, A305939 (analog for 9^k).

A305947(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(7^k))==n)

A305947_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(7^k)),nMax)]++);a[^-1]}

A305946 Number of powers of 6 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

14, 10, 17, 16, 11, 14, 10, 8, 12, 19, 9, 16, 13, 11, 10, 10, 11, 10, 10, 17, 7, 15, 14, 16, 13, 22, 12, 17, 15, 17, 7, 6, 14, 22, 13, 19, 14, 12, 15, 7, 11, 14, 6, 12, 9, 12, 9, 14, 13, 15, 21

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 14 is the number of terms in A030702 and in A195948, which includes the power 6^0 = 1.

These are the row lengths of A305926. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030702 = row 0 of A305926: k such that 6^k has no 0's; A238936: these powers 6^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063596 = column 1 of A305926: least k such that 6^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305946(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(6^k))==n)

A305946_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(6^k)),nMax)]++);a[^-1]}

A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, 25, 22, 20, 18, 21, 19, 19, 18, 24, 33, 17, 17, 18, 17, 14, 21, 26, 25, 23, 24, 29, 17, 22, 18, 21, 27, 26, 20, 21, 13, 27, 24, 12, 18, 24, 16, 17, 15, 30, 24, 32, 24, 12, 16, 16, 23, 23, 20, 23, 19, 23, 10, 21, 20, 21, 23, 20, 19, 23, 23, 22, 16, 18, 20, 20, 13, 15, 25, 24, 28, 24, 21, 16, 14, 23, 21, 19, 23, 19, 27, 26, 22, 18, 27, 16, 31, 21, 18, 25, 24

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.

These are the row lengths of A305933. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishingly small, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.
Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305943(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(3^k))==n)

A305943_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(3^k)),nMax)]++);a[^-1]}

A305944 Number of powers of 4 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

16, 22, 17, 14, 11, 19, 15, 15, 21, 20, 17, 22, 12, 13, 17, 24, 16, 19, 8, 17, 11, 15, 17, 15, 20, 17, 18, 20, 17, 21, 16, 19, 16, 14, 15, 19, 20, 24, 7, 16, 13, 14, 13, 14, 22, 22, 15, 18, 16, 16, 25

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 16 is the number of terms in A030701 and in A238940, which includes the power 4^0 = 1.

These are the row lengths of A305924. It remains an open problem to provide a proof that these rows are complete (as are all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030701 = row 0 of A305924: k such that 4^k has no 0's; A238940: these powers 4^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063575 = column 1 of A305924: least k such that 4^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305944(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(4^k))==n)

A305944_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(4^k)),nMax)]++);a[^-1]}

A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7, 19, 13, 15, 13, 10, 15, 12, 15, 11, 11, 15, 10, 9, 15, 17, 16, 13, 12, 12, 11, 14, 9, 14, 15, 16, 14, 13, 14, 15, 24, 14

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 16 is the number of terms in A008839 and in A195948, which includes the power 5^0 = 1.

These are the row lengths of A305925. It remains an open problem to provide a proof that these rows are complete (as are all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030701 (= row 0 of A305925): k such that 5^k has no 0's; A195948: these powers 4^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063585 (= column 1 of A305925): least k such that 5^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305945(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(5^k))==n)

A305945_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(5^k)),nMax)]++);a[^-1]}

A306118 Largest k such that 8^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

27, 43, 77, 61, 69, 119, 115, 158, 159, 168, 216, 232, 202, 198, 244, 270, 229, 274, 241, 273, 364, 283, 413, 298, 408, 341, 378, 431, 404, 403, 465, 483, 472, 454, 467, 508, 540, 575, 485, 576, 511, 623, 538, 515, 560, 655, 647, 661, 648, 639, 752

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030704: exponents of powers of 8 without digit 0 in base 10.

There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A063616: least k such that 8^k has n digits 0 in base 10.
Cf. A305938: number of k's such that 8^k has n digits 0.
Cf. A305928: row n lists exponents of 8^k with n digits 0.
Cf. A030704: { k | 8^k has no digit 0 } : row 0 of the above.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A071531: least k such that n^k contains a digit 0 in base 10.
Cf. A103663: least x such that x^n has no digit 0 in base 10.
Cf. A306112, ..., A306119: analog for 2^k, ..., 9^k.

A306118_vec(nMax,M=99*nMax+199,x=8,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

cp's OEIS Frontend

A071531 Smallest exponent r such that n^r contains at least one zero digit (in base 10).

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A305939 Number of powers of 9 having exactly n digits '0' (in base 10), conjectured.

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A305947 Number of powers of 7 having exactly n digits '0' (in base 10), conjectured.

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A305946 Number of powers of 6 having exactly n digits '0' (in base 10), conjectured.

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A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

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A305944 Number of powers of 4 having exactly n digits '0' (in base 10), conjectured.

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A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

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A306118 Largest k such that 8^k has exactly n digits 0 (in base 10), conjectured.

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