cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			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, ...

Links

Crossrefs

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).

Programs

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

Extensions

More terms from Erich Friedman
Definition clarified by Joerg Arndt, Sep 27 2016

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

Views

Author

Robert G. Wilson v, Aug 10 2001

Keywords

Links

Crossrefs

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).

Programs

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

Extensions

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

Views

Author

Robert G. Wilson v, Aug 10 2001

Keywords

Crossrefs

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).

Programs

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

Extensions

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

A305938 Number of powers of 8 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

14, 11, 15, 11, 6, 12, 10, 7, 14, 21, 9, 9, 15, 8, 6, 10, 8, 13, 11, 13, 7, 10, 12, 8, 16, 10, 10, 10, 9, 14, 18, 11, 15, 12, 9, 9, 10, 17, 8, 12, 8, 12, 9, 8, 8, 12, 10, 17, 12, 6, 16
Offset: 0

Views

Author

M. F. Hasler, Jun 22 2018

Keywords

Comments

a(0) = 14 is the number of terms in A030704 and in A195946, which includes the power 7^0 = 1.
These are the row lengths of A305928. 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.

Links

  • 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.)

Crossrefs

Cf. A030704 (= row 0 of A305928): k such that 8^k has no 0's; A195946: these powers 8^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063616 (= column 1 of A305928): least k such that 8^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305939 (analog for 9^k).

Programs

  • PARI
    A305947(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(8^k))==n)
  • PARI
    A305947_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(8^k)),nMax)]++);a[^-1]}

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

Views

Author

Robert G. Wilson v, Aug 10 2001

Keywords

Crossrefs

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).

Programs

  • Mathematica
    a = {}; Do[k = 0; While[ Count[ IntegerDigits[9^k], 0] != n, k++ ]; a = Append[a, k], {n, 0, 50} ]; a
  • PARI
    A063626(n)=for(k=0, oo, #select(d->!d, digits(9^k))==n&&return(k)) \\ M. F. Hasler, Jun 15 2018

Extensions

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

A306118 Largest k such that 8^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

27, 43, 77, 61, 69, 119, 115, 158, 159, 168, 216, 232, 202, 198, 244, 270, 229, 274, 241, 273, 364, 283, 413, 298, 408, 341, 378, 431, 404, 403, 465, 483, 472, 454, 467, 508, 540, 575, 485, 576, 511, 623, 538, 515, 560, 655, 647, 661, 648, 639, 752
Offset: 0

Views

Author

M. F. Hasler, Jun 22 2018

Keywords

Comments

a(0) is the largest term in A030704: exponents of powers of 8 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.

Links

  • M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 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.)

Crossrefs

Cf. A063616: least k such that 8^k has n digits 0 in base 10.
Cf. A305938: number of k's such that 8^k has n digits 0.
Cf. A305928: row n lists exponents of 8^k with n digits 0.
Cf. A030704: { k | 8^k has no digit 0 } : row 0 of the above.
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.

Programs

  • PARI
    A306118_vec(nMax,M=99*nMax+199,x=8,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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