A031146 -id:A031146 - cp's OEIS frontend

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

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

A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, 1091, 1239, 1201, 1224, 1210, 1141, 1339, 1240, 1282, 1395, 1449, 1416, 1408, 1616, 1524, 1727, 1725, 1553, 1942, 1907, 1945, 1870, 1724, 1972, 1965, 2075, 1983, 2114, 2257, 2256

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A007377: exponents of powers of 2 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. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A305942: number of k's such that 2^k has n digits 0.
Cf. A305932: row n lists exponents of 2^k with n digits 0.
Cf. A007377: { k | 2^k has no digit 0 } : row 0 of the above.
Cf. A238938: { 2^k having no digit 0 }.
Cf. A027870: number of 0's in 2^n (and A065712, A065710, A065714, A065715, A065716, A065717, A065718, A065719, A065744 for digits 1 .. 9).
Cf. A102483: 2^n contains no 0 in base 3.

A306112_vec(nMax,M=99*nMax+199,x=2,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

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

A031147 Smallest power of 2 containing exactly n zeros.

Original entry on oeis.org

1, 1024, 4398046511104, 8796093022208, 604462909807314587353088, 309485009821345068724781056, 1267650600228229401496703205376, 5070602400912917605986812821504, 784637716923335095479473677900958302012794430558004314112

Offset: 0

N. J. A. Sloane

Cf. A031146.

Table[SelectFirst[{#,DigitCount[#,10,0]}&/@(2^Range[0,2000]),#[[2]]==n&],{n,0,8}][[;;,1]] (* Harvey P. Dale, Sep 10 2024 *)

A031147(n)=for(k=n, oo, #select(d->!d, digits(2^k))==n&&return(2^k)) \\ M. F. Hasler, Jun 15 2018

a(n) = 2^A031146(n). - M. F. Hasler, Jun 15 2018 (corrected by Sean A. Irvine, Apr 11 2020)

More terms from Erich Friedman

a(8) added by M. F. Hasler, Jun 16 2018

A330790 Numbers k where 2^k in decimal has a record number of zeros.

Original entry on oeis.org

0, 10, 42, 43, 79, 88, 100, 102, 189, 198, 242, 250, 252, 262, 306, 368, 383, 447, 464, 466, 557, 628, 654, 657, 746, 771, 798, 905, 988, 989, 1044, 1114, 1118, 1217, 1316, 1461, 1486, 1493, 1703, 1705, 1753, 1926, 1993, 2122, 2159, 2368, 2462, 2502, 2594, 2680, 2739, 2900, 2936

Offset: 1

Chai Wah Wu, Feb 11 2020

Where records occur in A027870. The numbers of zeros are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 19, 20, 21, 24, 25, 28, ...

Chai Wah Wu, Table of n, a(n) for n = 1..2741

Cf. A027870, A031146, A031147.

cp's OEIS Frontend

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

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A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

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

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

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Extensions

A031147 Smallest power of 2 containing exactly n zeros.

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Mathematica

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Formula

Extensions

A330790 Numbers k where 2^k in decimal has a record number of zeros.

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