A238940 -id:A238940 - cp's OEIS frontend

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

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]}

A252482 Exponents n such that the decimal expansion of the power 12^n contains no zeros.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 14, 20, 26

Offset: 1

M. F. Hasler, Dec 17 2014

Conjectured to be finite.

See A245853 for the actual powers 12^a(n).

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014
Eric Weisstein's World of Mathematics, Zero

For zeroless powers x^n, see A238938 (x=2), A238939, A238940, A195948, A238936, A195908, A245852, A240945 (k=9), A195946 (x=11), A245853, A195945; A195942, A195943, A103662.
For the corresponding exponents, see A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, this sequence A252482, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[Range[0,30],DigitCount[12^#,10,0]==0&] (* Harvey P. Dale, Apr 06 2019 *)

for(n=0,9e9,vecmin(digits(12^n))&&print1(n","))

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]}

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

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A252482 Exponents n such that the decimal expansion of the power 12^n contains no zeros.

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

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PARI