A030702 -id:A030702 - cp's OEIS frontend

A239011 Exponents m such that the decimal expansion of 6^m exhibits its first zero from the right later than any previous exponent.

0, 2, 3, 4, 6, 7, 8, 12, 17, 24, 29, 42, 44, 101, 104, 128, 1015, 1108, 2629, 9683, 676076, 917474, 34882222, 53229360, 58230015, 90064345, 309000041, 319582553, 342860474, 382090917, 2770253437, 4380407969, 4407585753, 6966554399, 21235488251, 99404304146

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030702.

Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.

Cf. A000400, A030702, A020665, A031142, A239008, A239009, A239010, A239012, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[6, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 10000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(27)-a(34) from Bert Dobbelaere, Jan 21 2019

a(35)-a(36) from Chai Wah Wu, Jan 23 2020

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]}

A305926 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 6^k has n digits '0' (conjectured).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44, 10, 11, 14, 15, 18, 22, 28, 40, 42, 59, 9, 16, 20, 21, 26, 30, 31, 33, 37, 38, 39, 45, 46, 49, 51, 53, 63, 13, 23, 25, 27, 32, 34, 35, 36, 47, 48, 54, 61, 72, 73, 76, 82, 19, 52, 60, 64, 65, 70, 71, 83, 91, 93, 98, 43, 50

Offset: 0

M. F. Hasler, Jun 19 2018

The set of (nonempty) rows forms a partition of the nonnegative integers.
Read as a flattened sequence, a permutation of the nonnegative integers.
In the same way, another choice of (basis, digit, base) = (m, d, b) different from (6, 0, 10) will yield a similar partition of the nonnegative integers, trivial if m is a multiple of b.
It remains an open problem to provide a proof that the rows are complete, in the same way as each of the terms of A020665 is unproved.
We can also decide that the rows are to be truncated as soon as no term is found within a sufficiently large search limit. (For all of the displayed rows, there is no additional term up to many orders of magnitude beyond the last term.) That way the rows are well-defined, but it is no longer guaranteed to have a partition of the integers.
The author finds this sequence "nice", i.e., appealing (as well as, e.g., the variant A305933 for basis 3) in view of the idea of partitioning the integers in such an elementary yet highly nontrivial way, and the remarkable fact that the rows are just roughly one line long. Will this property remain for large n, or else, how will the row lengths evolve?

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44 (= A030702)
1 : 10, 11, 14, 15, 18, 22, 28, 40, 42, 59
2 : 9, 16, 20, 21, 26, 30, 31, 33, 37, 38, 39, 45, 46, 49, 51, 53, 63
3 : 13, 23, 25, 27, 32, 34, 35, 36, 47, 48, 54, 61, 72, 73, 76, 82
4 : 19, 52, 60, 64, 65, 70, 71, 83, 91, 93, 98
5 : 43, 50, 55, 58, 62, 66, 67, 75, 77, 78, 101, 106, 129, 134
...
Column 0 is A063596: least k such that 6^k has n digits '0' in base 10.
Row lengths are 14, 10, 17, 16, 11, 14, 10, 8, 12, 19, 9, 16, 13, 11, 10, 10, 11, 10, 10, 17, ... (A305946).
Last terms of the rows yield (44, 59, 63, 82, 98, 134, 108, 123, 199, 189, 192, 200, 275, 282, 267, 307, 298, 296, 391, 338, ...), A306116.
The inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 24, 14, 15, 9, 41, 16, 17, 25, 10, 18, 57, 26, 27, 19, 42, 11, 43, 28, 44, 20, 12, 29, 30, ...), not in OEIS.

M. F. Hasler, Zeroless powers.. OEIS Wiki, March 2014.

Cf. A030702, A063596.

Cf. A305932 (analog for 2^k), A305933 (analog for 3^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).

mx = 1000; g[n_] := g[n] = DigitCount[6^n, 10, 0]; f[n_] := Select[Range@mx, g@# == n &]; Table[f@n, {n, 0, 4}] // Flatten (* Robert G. Wilson v, Jun 20 2018 *)

apply( A305926_row(n,M=50*(n+1))=select(k->#select(d->!d,digits(6^k))==n,[0..M]), [0..19])

A195985 Least prime such that p^2 is a zeroless n-digit number.

Original entry on oeis.org

2, 5, 11, 37, 107, 337, 1061, 3343, 10559, 33343, 105517, 333337, 1054133, 3333373, 10540931, 33333359, 105409309, 333333361, 1054092869, 3333333413, 10540925639, 33333333343, 105409255363, 333333333367, 1054092553583, 3333333333383, 10540925534207

Offset: 1

M. F. Hasler, Sep 26 2011

			a(1)^2=4, a(2)^2=25, a(3)^2=121, a(4)^2=1369 are the least squares of primes with 1, 2, 3 resp. 4 digits, and these digits are all nonzero.
a(5)=107 since 101^2=10201 and 103^2=10609 both contain a zero digit, but 107^2=11449 does not.
a(1000)=[10^500/3]+10210 (500 digits), since primes below sqrt(10^999) = 10^499*sqrt(10) ~ 3.162e499 have squares of less than 1000 digits, between sqrt(10^999) and 10^500/3 = sqrt(10^1000/9) ~ 3.333...e499 they have at least one zero digit. Finally, the 7 primes between 10^500/3 and a(1000) also happen to have a "0" digit in their square, but not so
  a(1000)^2 = 11111...11111791755555...55555659792849
  = [10^500/9]*(10^500+5) + 6806*10^500+104237294.

M. F. Hasler, Table of n, a(n) for n = 1..158
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), as of Jan 30 2003.

Cf. A195942, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a(n)={ my(p=sqrtint(10^n\9)-1); until( is_A052382(p^2), p=nextprime(p+2));p}

A050727 Numbers k such that the decimal expansion of 6^k contains no pair of consecutive equal digits (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 11, 13, 14, 15, 26

Offset: 1

Patrick De Geest, Sep 15 1999

No additional terms up to 25000. - Harvey P. Dale, Oct 17 2011
No additional terms up to 100000. - Michel Marcus, Oct 16 2019
No additional terms up to 10^7. - Lucas A. Brown, Mar 02 2024

			6^26 = 170581728179578208256 where no consecutive digits are equal.

Cf. A000400, A030702, A046264, A046272.

Select[Range[120],!MemberQ[Differences[IntegerDigits[6^#]],0]&] (* Harvey P. Dale, Oct 17 2011 *)

isok(n) = {my(d = digits(6^n), c = d[1]); for (i=2, #d, if (d[i] == c, return (0)); c = d[i];); return (1);} \\ Michel Marcus, Oct 16 2019

try: from gmpy2 import mpz; x = mpz(1)
except: x = 1
print(0)
k = 1
while True:
    print('\b'*42 + str(k), end='')
    x *= 6  # x == 6**k
    y, flag = x, True
    y, a = divmod(y, 10)
    while y > 6:
        b = a
        y, a = divmod(y, 10)
        if a == b:
            flag = False
            break
    if flag: print()
    k += 1
# Lucas A. Brown, Mar 02 2024

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","))

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]}

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A239011 Exponents m such that the decimal expansion of 6^m exhibits its first zero from the right later than any previous exponent.

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

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A305926 Irregular table: row n >= 0 lists all k >= 0 such that the decimal representation of 6^k has n digits '0' (conjectured).

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A195985 Least prime such that p^2 is a zeroless n-digit number.

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A050727 Numbers k such that the decimal expansion of 6^k contains no pair of consecutive equal digits (probably finite).

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

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

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