A238939 -id:A238939 - cp's OEIS frontend

A305934 Powers of 3 that have exactly one digit '0' in base 10.

59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 847288609443, 68630377364883, 328256967394537077627, 26588814358957503287787, 717897987691852588770249, 6461081889226673298932241, 1144561273430837494885949696427, 22528399544939174411840147874772641, 67585198634817523235520443624317923

Offset: 1

M. F. Hasler (following a suggestion by Zak Seidov), Jun 14 2018

Motivated by A030700: decimal expansion of 3^n contains no zeros (probably finite).
It appears that this sequence is finite. Is a(15) = 3^73 the last term?
There are no more terms through at least 3^(10^7) (which is a 4771213-digit number). It seems nearly certain that no power of 3 containing this many or more decimal digits could have fewer than two '0' digits. (Among numbers of the form 3^k with 73 < k <= 10^7, the only one having fewer than two '0' digits among its final 200 digits is 3^5028978.) - Jon E. Schoenfield, Jun 24 2018
The first 6 terms coincide with A305931: powers of 3 having at least one digit 0, with complement A238939 (within A000244: powers of 3) conjectured to be finite, too. Then, a(7..8) = A305931(9..10), etc.

Cf. A030700: decimal expansion of 3^n contains no zeros (probably finite), A238939: powers of 3 with no digit '0' in their decimal expansion, A000244: powers of 3.
Subsequence of A305931: powers of 3 having at least one '0'.
Cf. A305933: row n = { k | 3^k has n digits '0' }.

Select[3^Range[120], DigitCount[#, 10, 0] == 1 &] (* Michael De Vlieger, Jul 01 2018 *)

for(n=1,99, #select(t->!t,digits(3^n))==1&& print1(3^n","))

a(n) = 3^A305933(1,n).

A305931 Powers of 3 having at least one digit '0' in their decimal representation.

Original entry on oeis.org

59049, 14348907, 43046721, 129140163, 387420489, 3486784401, 10460353203, 31381059609, 847288609443, 68630377364883, 205891132094649, 1853020188851841, 5559060566555523, 50031545098999707, 150094635296999121, 450283905890997363, 1350851717672992089, 4052555153018976267, 12157665459056928801

Offset: 1

M. F. Hasler, Jun 15 2018

The analog of A298607 for 3^k instead of 2^k.

The complement A238939 is conjectured to have only 23 elements, the largest being 3^68. Thus, all larger powers of 3 are (conjectured to be) in this sequence. Each of the subsequences "powers of 3 with exactly n digits 0" is conjectured to be finite. Provided there is at least one such element for each n >= 0, this leads to a partition of the integers, given in A305933.

Cf. A030700 = row 0 of A305933: decimal expansion of 3^n contains no zeros.
Complement (within A000244: powers of 3) of A238939: powers of 3 with no digit '0' in their decimal expansion.
Analog of A298607: powers of 2 with the digit '0' in their decimal expansion.
The first six terms coincide with the finite sequence A305934: powers of 3 having exactly one digit 0.

Select[3^Range[0,40],DigitCount[#,10,0]>0&] (* Harvey P. Dale, May 30 2020 *)

for(k=0,69, vecmin(digits(3^k))|| print1(3^k","))

select( t->!vecmin(digits(t)), apply( k->3^k, [0..40]))

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]}

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","))

A306113 Largest k such that 3^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

68, 73, 136, 129, 205, 237, 317, 268, 251, 276, 343, 372, 389, 419, 565, 416, 494, 571, 637, 628, 713, 629, 638, 655, 735, 690, 862, 802, 750, 863, 826, 996, 976, 1008, 1085, 1026, 1130, 995, 962, 1082, 1136, 1064, 1176, 1084, 1215, 1354, 1298, 1275, 1226, 1468, 1353

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030700: exponents of powers of 3 without digit 0.

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. A063555: least k such that 3^k has n digits 0 in base 10.
Cf. A305943: number of k's such that 3^k has n digits 0.
Cf. A305933: row n lists exponents of 3^k with n digits 0.
Cf. A030700: { k | 3^k has no digit 0 } : row 0 of the above.
Cf. A238939: { 3^k having no digit 0 }.
Cf. A305930: number of 0's in 3^n.
Cf. A306112, ..., A306119: analog for 2^k, ..., 9^k.

A306113_vec(nMax,M=99*nMax+199,x=3,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A358271 Product of the digits of 3^n.

Original entry on oeis.org

1, 3, 9, 14, 8, 24, 126, 112, 180, 1296, 0, 1372, 240, 3240, 217728, 0, 0, 0, 0, 24192, 0, 0, 0, 2709504, 6635520, 0, 66355200, 8534937600, 731566080, 0, 0, 10369949184, 0, 0, 399983754240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6243870843076608000

Offset: 0

Joseph Caliendo, Nov 06 2022

a(68) is likely the last nonzero term; see A030700 and A238939. - Michael S. Branicky, Nov 06 2022

			For a(0), 3^0 = 1 with product of digits 1;
for a(3), 3^3 = 27 with product of digits 2*7 = 14;
for a(10), 3^10 = 59049 with product of digits 5*9*0*4*9 = 0.

Cf. A007954, A000244.
Cf. A030700, A238939.
Cf. A014257.

a[n_] := Times @@ IntegerDigits[3^n]; Array[a, 69, 0] (* Amiram Eldar, Nov 07 2022 *)

a(n) = vecprod(digits(3^n)); \\ Michel Marcus, Nov 07 2022

from math import prod
def a(n): return prod(map(int, str(3**n)))
print([a(n) for n in range(69)]) # Michael S. Branicky, Nov 06 2022

a(n) = A007954(A000244(n)).

More terms from Michael S. Branicky, Nov 06 2022

cp's OEIS Frontend

A305934 Powers of 3 that have exactly one digit '0' in base 10.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A305931 Powers of 3 having at least one digit '0' in their decimal representation.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A306113 Largest k such that 3^k has exactly n digits 0 (in base 10), conjectured.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A358271 Product of the digits of 3^n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions