A063555 -id:A063555 - cp's OEIS frontend

A031146 Exponent of the least power of 2 having exactly n zeros in its decimal representation.

0, 10, 42, 43, 79, 88, 100, 102, 189, 198, 242, 250, 252, 263, 305, 262, 370, 306, 368, 383, 447, 464, 496, 672, 466, 557, 630, 629, 628, 654, 657, 746, 771, 798, 908, 913, 917, 906, 905, 1012, 1113, 988, 1020, 989, 1044, 1114, 1120, 1118, 1221, 1218, 1255

Offset: 0

N. J. A. Sloane

			a(3) = 43 since 2^m contains 3 0's for m starting with 43 (2^43 = 8796093022208) and followed by 53, 61, 69, 70, 83, 87, 89, 90, 93, ...

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000079, A007377, A031147, A006889.

Cf. A063555 (analog for 3^k), A063575 (for 4^k), A063585 (for 5^k), A063596 (for 6^k), A063606 (for 7^k), A063616 (for 8^k), A063626 (for 9^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[2^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a (* Robert G. Wilson v, Jun 12 2004 *)
nn = 100; t = Table[0, {nn}]; found = 0; k = 0; While[found < nn, k++; cnt = Count[IntegerDigits[2^k], 0]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = k; found++]]; t = Join[{0}, t] (* T. D. Noe, Mar 14 2012 *)

A031146(n)=for(k=0, oo, #select(d->!d, digits(2^k))==n&&return(k)) \\ M. F. Hasler, Jun 15 2018

More terms from Erich Friedman

Definition clarified by Joerg Arndt, Sep 27 2016

A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68, 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73, 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136, 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129

Offset: 0

M. F. Hasler, Jun 14 2018

The set of nonempty rows is 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 (3, 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, just 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 we are no longer guaranteed to get a partition of the integers.
The author finds the idea of partitioning the integers in this elementary yet highly nontrivial way appealing, as is the fact that the initial rows are just roughly one line long. Will this property continue to hold for large n, or if not, how will the row lengths evolve?

			The table reads:
n \ k's
0 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, 31, 34, 68 (cf. A030700)
1 : 10, 15, 16, 17, 18, 20, 25, 29, 43, 47, 50, 52, 63, 72, 73
2 : 22, 30, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 51, 53, 56, 58, 60, 61, 62, 64, 69, 71, 83, 93, 96, 108, 111, 123, 136
3 : 21, 37, 49, 67, 75, 81, 82, 87, 90, 105, 112, 121, 129
4 : 35, 59, 65, 66, 70, 74, 77, 79, 88, 98, 106, 116, 117, 128, 130, 131, 197, 205
5 : 57, 76, 78, 80, 86, 89, 91, 92, 101, 102, 104, 109, 115, 118, 122, 127, 134, 135, 164, 166, 203, 212, 237
...
The first column is A063555: least k such that 3^k has n digits '0' in base 10.
Row lengths are 23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, ... (A305943).
Last term of the rows (i.e., largest k such that 3^k has exactly n digits 0) are (68, 73, 136, 129, 205, 237, 317, 268, 251, 276, 343, ...), A306113.
Inverse permutation is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 10, 11, 12, 13, 24, 25, 26, 27, 14, 28, 69, 38, 15, 16, 29, 17, 18, 19, 30, 39, 20, ...), not in OEIS.

Cf. A030700, A063555.
Cf. A305932 (analog for 2^k), A305924 (analog for 4^k), ..., A305929 (analog for 9^k).
Cf. A305934: powers of 3 with exactly one '0', A305943: powers of 3 with at least one '0'.

apply( A305933_row(n,M=50*n+70)=select(k->#select(d->!d,digits(3^k))==n,[0..M]), [0..10])
print(apply(t->#t,%)"\n"apply(vecmax,%)"\n"apply(t->t-1,Vec(vecsort(concat(%),,1)[1..99]))) \\ to show row lengths, last elements, and inverse permutation.

A063575 Smallest k such that 4^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 5, 21, 35, 47, 44, 50, 51, 103, 99, 121, 125, 126, 175, 166, 131, 185, 153, 184, 223, 272, 232, 248, 336, 233, 306, 315, 384, 314, 327, 333, 373, 393, 399, 454, 457, 504, 453, 484, 506, 621, 494, 510, 639, 522, 557, 560, 559, 716, 609, 629

Offset: 0

Robert G. Wilson v, Aug 10 2001

Harvey P. Dale, Table of n, a(n) for n = 0..1000 (first 150 terms from Harry J. Smith; a(0) modified by M. F. Hasler).

Cf. A031146 (analog for 2^k), A063555 (for 3^k), A063585 (for 5^k), A063596 (for 6^k), A063606 (for 7^k), A063616 (for 8^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[4^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a
Module[{nn=750,p4},p4=Table[{n,DigitCount[4^n,10,0]},{n,nn}];Transpose[ Table[ SelectFirst[p4,#[[2]]==i&],{i,0,50}]][[1]]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, May 20 2016 *)

Count(x, d)= { local(c,f); c=0; while (x>9, f=x-10*(x\10); if (f==d, c++); x\=10); if (x==d, c++); return(c) } { for (n=0, 150, a=0; while (Count(4^a, 0) != n, a++); write("b063575.txt", n, " ", a) ) } \\  Harry J. Smith, Aug 26 2009

A063575(n)=for(k=n,oo,#select(d->!d,digits(4^k))==n&&return(k)) \\ M. F. Hasler, Jun 14 2018

a(0) changed to 0 as in A031146, A063555, ... by M. F. Hasler, Jun 14 2018

A063596 Least k >= 0 such that 6^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 10, 9, 13, 19, 43, 56, 41, 94, 79, 113, 100, 88, 112, 124, 127, 138, 176, 144, 175, 174, 168, 170, 210, 245, 228, 182, 237, 287, 260, 312, 321, 294, 347, 389, 365, 401, 386, 390, 419, 460, 425, 438, 426, 488, 490, 520, 458, 489, 521, 513

Offset: 0

Robert G. Wilson v, Aug 10 2001

Cf. A031146 (analog for 2^k), A063555 (for 3^k), A063575 (for 4^k), A063585 (for 5^k), A063606 (for 7^k), A063616 (for 8^k), A063626 (for 9^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[6^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a
With[{pwr6=Table[{n,DigitCount[6^n,10,0]},{n,1000}]},Join[{0},Transpose[ Table[ SelectFirst[pwr6,#[[2]]==i&],{i,60}]][[1]]]] (* Harvey P. Dale, Dec 15 2014 *)

A063596(n)=for(k=0, oo, #select(d->!d, digits(6^k))==n&&return(k)) \\ M. F. Hasler, Jun 14 2018

a(0) changed to 0 (as in A031146, A063555, ...) and better title from M. F. Hasler, Jun 14 2018

A063585 Least k >= 0 such that 5^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 8, 13, 34, 40, 48, 52, 45, 64, 99, 143, 132, 100, 122, 117, 151, 205, 207, 201, 242, 230, 244, 231, 221, 295, 264, 266, 333, 248, 344, 346, 274, 391, 345, 356, 393, 433, 365, 472, 499, 488, 455, 516, 485, 511, 458, 520, 487, 459, 456, 457

Offset: 0

Robert G. Wilson v, Aug 10 2001

Robert Israel, Table of n, a(n) for n = 0..3373
M. F. Hasler, Zeroless powers. OEIS Wiki, March 2014

Cf. A031146 (analog for 2^k), A063555 (analog for 3^k), A063575 (analog for 4^k), A063596 (analog for 6^k).

N:= 100: # to get a(0)..a(N)
A:= Array(0..N, -1):
p:= 1: A[0]:= 0:
count:= 1:
for k from 1 while count <= N do
  p:= 5*p;
  m:= numboccur(0, convert(p, base, 10));
  if m <= N and A[m] < 0 then A[m]:= k; count:= count+1;
od:
convert(A,list); # Robert Israel, Dec 20 2018

a = {}; Do[k = 0; While[ Count[ IntegerDigits[5^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a

A063585(n)=for(k=n,oo,#select(d->!d,digits(5^k))==n&&return(k)) \\ M. F. Hasler, Jun 14 2018

a(0) changed to 0 (as in A031146, A063555, ...) and better title from M. F. Hasler, Jun 14 2018

A063606 Smallest k >= 0 such that 7^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 4, 9, 13, 25, 55, 39, 41, 45, 70, 69, 65, 75, 107, 109, 134, 167, 142, 156, 196, 157, 205, 214, 180, 213, 183, 162, 251, 263, 276, 268, 290, 306, 295, 369, 313, 332, 293, 353, 340, 357, 387, 367, 476, 334, 509, 363, 474, 454, 488, 453

Offset: 0

Robert G. Wilson v, Aug 10 2001

Cf. A031146 (analog for 2^k), A063555 (analog for 3^k), A063575 (analog for 4^k), A063585 (for 5^k), A063596 (analog for 6^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[7^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a
Module[{p7=DigitCount[#,10,0]&/@(7^Range[600]),nn=60},Join[{0},Flatten[ Table[ Position[p7,n,1,1],{n,nn}]]]] (* Harvey P. Dale, Apr 12 2022 *)

A063606(n)=for(k=n, oo, #select(d->!d, digits(5^k))==n&&return(k)) \\ M. F. Hasler, Jun 14 2018

A063616 Smallest k >= 0 such that 8^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 4, 14, 23, 42, 33, 35, 34, 63, 66, 87, 116, 84, 101, 126, 164, 128, 102, 135, 143, 149, 155, 203, 224, 186, 204, 210, 237, 261, 218, 219, 286, 257, 266, 361, 355, 336, 302, 374, 339, 371, 398, 340, 409, 348, 388, 494, 436, 407, 406, 439

Offset: 0

Robert G. Wilson v, Aug 10 2001

Cf. A031146 (analog for 2^k), A063555 (analog for 3^k), A063575 (analog for 4^k), A063585 (for 5^k), A063596 (analog for 6^k), A063606 (analog for 7^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[8^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a

A063616(n)=for(k=0, oo, #select(d->!d, digits(8^k))==n&&return(k)) \\ M. F. Hasler, Jun 14 2018

a(0) changed to 0 (as in A031146, A063555, ...) and better title from M. F. Hasler, Jun 14 2018

A063626 Smallest k >= 0 such that 9^k has exactly n 0's in its decimal representation.

Original entry on oeis.org

0, 5, 11, 41, 33, 38, 42, 27, 60, 71, 63, 85, 94, 139, 96, 127, 157, 166, 131, 160, 170, 148, 190, 210, 212, 203, 221, 222, 218, 257, 223, 243, 250, 275, 302, 255, 273, 271, 333, 372, 270, 339, 371, 457, 408, 347, 402, 410, 483, 448, 355

Offset: 0

Robert G. Wilson v, Aug 10 2001

Cf. A031146 (analog for 2^k), A063555 (for 3^k), A063575 (for 4^k), A063585 (for 5^k), A063596 (for 6^k), A063606 (for 7^k), A063616 (for 8^k).

a = {}; Do[k = 0; While[ Count[ IntegerDigits[9^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a

A063626(n)=for(k=0, oo, #select(d->!d, digits(9^k))==n&&return(k)) \\ M. F. Hasler, Jun 15 2018

a(0) changed to 0 (as in A031146, A063555, ...) and better title from M. F. Hasler, Jun 15 2018

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]}

A305930 Number of digits '0' in 3^n (in base 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 3, 2, 0, 0, 1, 0, 0, 0, 1, 2, 0, 2, 2, 0, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 3, 1, 2, 1, 2, 7, 6, 2, 5, 2, 4, 2, 2, 2, 1, 2, 4, 4, 3, 0, 2, 4, 2, 1, 1, 4, 3, 5, 4, 5, 4, 5, 3, 3, 2, 6, 6, 5, 3, 4, 5, 3, 5, 5, 2, 6, 6, 2, 6, 4, 7

Offset: 0

M. F. Hasler, Jun 22 2018

			3^10 = 59049 is the smallest power of 3 having a digit 0, so a(10) = 1 is the first nonzero term.

Cf. A027870 (analog for 2^k), A030700 (indices of zeros).
Cf. A063555: index of first appearence of n in this sequence.
Cf. A305933: table with n in row a(n).

Haskell
```
a305930 = a055641 . a000244
```

Table[ Count[ IntegerDigits[3^n], 0], {n, 0, 100} ]
DigitCount[3^Range[0,110],10,0]

apply( A305930(n)=#select(d->!d,digits(3^n)), [0..99])

a(n) = A055641(A000244(n)).

a(A030700(n)) = 0; a(A305934(n)) = 1; a(A305931(n)) >= 1; a(A305933(n,k)) = n.

cp's OEIS Frontend

A031146 Exponent of the least power of 2 having exactly n zeros in its decimal representation.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A305933 Irregular table read by rows: row n >= 0 lists all k >= 0 such that the decimal representation of 3^k has n digits '0' (conjectured).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A063575 Smallest k such that 4^k has exactly n 0's in its decimal representation.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A063596 Least k >= 0 such that 6^k has exactly n 0's in its decimal representation.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A063585 Least k >= 0 such that 5^k has exactly n 0's in its decimal representation.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A063606 Smallest k >= 0 such that 7^k has exactly n 0's in its decimal representation.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A063616 Smallest k >= 0 such that 8^k has exactly n 0's in its decimal representation.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A063626 Smallest k >= 0 such that 9^k has exactly n 0's in its decimal representation.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views