A071531 -id:A071531 - cp's OEIS frontend

A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.

36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, 35, 29, 27, 27, 39, 34, 34, 28, 29, 31, 30, 38, 25, 35, 35, 36, 40, 32, 40, 43, 39, 32, 30, 30, 32, 36, 39, 23, 26, 31, 37, 27, 28, 33, 39, 28, 44, 34, 27, 43, 33, 27, 32, 31, 27, 27, 32, 35, 34, 36, 28, 32, 39, 38, 40, 28, 43, 38, 32, 22

Offset: 0

M. F. Hasler, Jun 21 2018

a(0) = 36 is the number of terms in A007377 and in A238938, which includes the power 2^0 = 1.
These are the row lengths of A305932. 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.
The average of the first 100000 terms is ~33.219 with a minimum of 12 and a maximum of 61. - Hans Havermann, Apr 26 2020

Hans Havermann, Table of n, a(n) for n = 0..10000
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.)

Row lengths of A305932 (row n = exponents of 2^k with n '0's).
Cf. A007377 = {k | 2^k has no digit 0}; A238938: powers of 2 with no digit 0.
Cf. A298607: powers of 2 with the digit '0' in their decimal expansion.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A071531: least r such that n^r has a digit 0, in base 10.
Cf. A306112: largest k such that 2^k has n digits 0, in base 10.

A305942(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(2^k))==n)

A305942_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(2^k)),nMax)]++);a[^-1]}

A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

34, 36, 68, 56, 65, 106, 144, 134, 119, 138, 154, 186, 194, 191, 219, 208, 247, 267, 199, 314, 292, 263, 319, 303, 307, 345, 431, 401, 375, 388, 413, 498, 488, 504, 465, 513, 565, 464, 481, 541, 568, 532, 588, 542, 600, 677, 649, 633, 613, 734, 627

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030705: exponents of powers of 9 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, 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. A063626: least k such that 9^k has n digits 0 in base 10.
Cf. A305939: number of k's such that 9^k has n digits 0.
Cf. A305929: row n lists exponents of 9^k with n digits 0.
Cf. A030705: { k | 9^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, ..., A306118: analog for 2^k, ..., 8^k.

A306119_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)]=k);a[^-1]}

Data corrected thanks to a remark by R. J. Mathar, by M. F. Hasler, Feb 11 2023

A305941 Smallest power of n having at least one digit 0 (when written in base 10).

Original entry on oeis.org

1024, 59049, 1024, 390625, 10077696, 2401, 4096, 59049, 10, 161051, 20736, 4826809, 105413504, 50625, 4096, 410338673, 104976, 130321, 20, 4084101, 10648, 148035889, 191102976, 390625, 308915776, 14348907, 17210368, 707281, 30, 887503681, 1024, 1089, 39304, 1500625, 60466176, 50653, 2085136, 90224199, 40

Offset: 2

M. F. Hasler, Jun 19 2018

One would have a(0) = 0 (= 0^1), but a(1) is undefined.

The similar sequence defined as "...exactly one digit 0..." would be ill-defined for all powers of 100, 1001, and others.

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A071531.

Table[Module[{k=1},While[DigitCount[n^k,10,0]==0,k++];n^k],{n,2,40}] (* Harvey P. Dale, Aug 18 2025 *)

apply( A305941(n)=for(k=1,oo, vecmin(digits(n^k))||return(n^k)), [2..20])

a(n) >= n, with equality iff n is in A011540 \ {0} = {10, 20, ..., 100, 101, ...}.

A263450 Smallest integer k>0 such that there is at least one zero in the decimal representation of prime(n)^k.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Oct 18 2015

Conjecture: there are an infinite number of ones in the sequence.
Corresponding values of prime(n)^k: 1024, 59049, 390625, 2401, 161051, 4826809, 410338673, 130321 (not yet in OEIS).
From Robert Israel, Oct 19 2015: (Start)
By Dirichlet's theorem there are infinitely many n for which prime(n) == 1 (mod 100), and these all have a(n) = 1.
All a(n) <= 20, since every x coprime to 10 has x^20 == 1 (mod 100). (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062584, A071531, A103662, A103663.

f:= proc(m) local k;
for k from 1 do
     if has(convert(m^k,base,10),0) then return k fi
   od
end proc:
seq(f(ithprime(i)), i=1..1000); # Robert Israel, Oct 19 2015

Reap[Do[p=Prime[n];k=1;While[Min[IntegerDigits[p^k]]>0,k++];Sow[k],{n,1,200}]][[2,1]]

a(n) = {p = prime(n); k = 1; while (vecmin(digits(p^k)), k++); k;} \\ Michel Marcus, Oct 21 2015

a(n) = A071531(prime(n)). - Michel Marcus, Oct 21 2015

A306114 Largest k such that 4^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

43, 92, 77, 88, 115, 171, 182, 238, 235, 308, 324, 348, 412, 317, 366, 445, 320, 424, 362, 448, 546, 423, 540, 545, 612, 605, 567, 571, 620, 641, 619, 700, 708, 704, 808, 762, 811, 744, 755, 971, 896, 900, 935, 862, 986, 954, 982, 956, 1057, 1037, 1128

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030701: exponents of powers of 4 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, 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. A063575: least k such that 4^k has n digits 0 in base 10.
Cf. A305944: number of k's such that 4^k has n digits 0.
Cf. A305924: row n lists exponents of 4^k with n digits 0.
Cf. A030701: { k | 4^k has no digit 0 } : row 0 of the above.
Cf. A238940: { 4^k having no digit 0 }.
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.

A306114_vec(nMax,M=99*nMax+199,x=4,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A306115 Largest k such that 5^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

58, 85, 107, 112, 127, 157, 155, 194, 198, 238, 323, 237, 218, 301, 303, 324, 339, 476, 321, 284, 496, 421, 475, 415, 537, 447, 494, 538, 531, 439, 473, 546, 587, 588, 642, 690, 769, 689, 687, 686, 757, 732, 683, 826, 733, 825, 833, 810, 827, 888, 966

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A008839: exponents of powers of 5 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, 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. A063585: least k such that 5^k has n digits 0 in base 10.
Cf. A305945: number of k's such that 5^k has n digits 0.
Cf. A305925: row n lists exponents of 5^k with n digits 0.
Cf. A008839: { k | 5^k has no digit 0 } : row 0 of the above.
Cf. A195948: { 5^k having no digit 0 }.
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.

A306115_vec(nMax,M=99*nMax+199,x=5,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A306116 Largest k such that 6^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

44, 59, 63, 82, 98, 134, 108, 123, 199, 189, 192, 200, 275, 282, 267, 307, 298, 296, 391, 338, 340, 396, 328, 436, 432, 478, 484, 615, 428, 529, 492, 515, 536, 523, 627, 665, 559, 592, 637, 560, 654, 674, 590, 653, 728, 791, 753, 781, 812, 783, 788

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030702: exponents of powers of 6 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, 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. A063596: least k such that 6^k has n digits 0 in base 10.
Cf. A305946: number of k's such that 6^k has n digits 0.
Cf. A305926: row n lists exponents of 6^k with n digits 0.
Cf. A030702: { k | 6^k has no digit 0 } : row 0 of the above.
Cf. A238936: { 6^k having no digit 0 }.
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.

A306116_vec(nMax,M=99*nMax+199,x=6,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A306117 Largest k such that 7^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

35, 51, 93, 58, 122, 74, 108, 131, 118, 152, 195, 192, 236, 184, 247, 243, 254, 286, 325, 292, 318, 336, 375, 393, 339, 431, 327, 433, 485, 447, 456, 455, 448, 492, 452, 507, 489, 541, 526, 605, 627, 706, 730, 628, 665, 660, 798, 715, 704, 633, 728

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030703: exponents of powers of 7 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, 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. A063606: least k such that 7^k has n digits 0 in base 10.
Cf. A305947: number of k's such that 7^k has n digits 0.
Cf. A305927: row n lists exponents of 6^k with n digits 0.
Cf. A030703: { k | 7^k has no digit 0 } : row 0 of the above.
Cf. A195908: { 7^k having no digit 0 }.
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.

A306117_vec(nMax,M=99*nMax+199,x=7,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);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

A305942 Number of powers of 2 having exactly n digits '0' (in base 10), conjectured.

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A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

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A305941 Smallest power of n having at least one digit 0 (when written in base 10).

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A263450 Smallest integer k>0 such that there is at least one zero in the decimal representation of prime(n)^k.

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A306114 Largest k such that 4^k has exactly n digits 0 (in base 10), conjectured.

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A306115 Largest k such that 5^k has exactly n digits 0 (in base 10), conjectured.

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A306116 Largest k such that 6^k has exactly n digits 0 (in base 10), conjectured.

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A306117 Largest k such that 7^k has exactly n digits 0 (in base 10), conjectured.

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