A007377 -id:A007377 - cp's OEIS frontend

A240963 Numbers n such that the decimal expansion of n^n contains no zeros.

1, 2, 3, 4, 5, 6, 7, 8, 22

Offset: 1

Anthony Sand, Aug 05 2014

A240962(a(n)) = 0.
The sequence seems to be finite; compare A007377 (n such that 2^n contains no 0).
For no reason I can fathom this is a subsequence of A053408. - Charles R Greathouse IV, Aug 21 2014

			1^1 = 1
2^2 = 4
3^3 = 27
4^4 = 256
5^5 = 3125
6^6 = 46656
7^7 = 823543
8^8 = 16777216
22^22 = 341427877364219557396646723584

Cf. A240962, A007377, A030700, A053408.

Select[Range[10^4], If[Count[IntegerDigits[#^#], 0] == 0, True] &] (* Michael De Vlieger, Aug 07 2014 *)

for(n=1,10^3,d=digits(n^n);if(vecmin(d),print1(n,", "))) \\ Derek Orr, Aug 05 2014

A298607 Powers of 2 with the digit '0' in their decimal expansion.

Original entry on oeis.org

1024, 2048, 4096, 131072, 1048576, 2097152, 4194304, 8388608, 67108864, 536870912, 1073741824, 274877906944, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 1125899906842624

Offset: 1

Alonso del Arte, Jan 22 2018

The complement, A238938, is conjectured to be finite. Furthermore, Khovanova (see link) believes 2^86 = 77371252455336267181195264 is the largest power of 2 not in this sequence.

			2^12 = 4096 contains one 0 in its decimal representation, hence 4096 is in the sequence.
2^13 = 8192 contains no 0's and is thus not in the sequence.

Tanya Khovanova, "86 Conjecture", Tanya Khovanova's Math Blog, February 15, 2011.

Cf. A007377, A027870.

Select[2^Range[0, 63], DigitCount[#, 10, 0] > 0 &]

lista(nn) = {for (n=0, nn, if (vecsearch(Set(digits(p=2^n)), 0), print1(p, ", ")););} \\ Michel Marcus, Mar 05 2018

A352040 a(n) is the least number k such that 2^k contains each of the 10 digits at least n times.

Original entry on oeis.org

68, 88, 119, 200, 209, 246, 291, 318, 396, 398, 443, 443, 495, 586, 592, 622, 646, 707, 758, 758, 813, 866, 875, 903, 923, 1001, 1022, 1022, 1105, 1111, 1111, 1231, 1243, 1245, 1327, 1342, 1419, 1453, 1453, 1454, 1534, 1536, 1537, 1626, 1676, 1699, 1699, 1763

Offset: 1

Amiram Eldar, Apr 16 2022

a(1)-a(8) were given in the solution to Problem 410 (Crux Mathematicorum, 1979), but a(2) = 170 is wrong.

a(1) was calculated by Rudolph Ondrejka in 1976.

			a(1) = 68 since 2^68 = 295147905179352825856 is the least power of 2 that contains all the 10 digits at least once.

Amiram Eldar, Table of n, a(n) for n = 1..10000
James Gary Propp, Problem 410, Crux Mathematicorum, Vol. 5, No. 1 (1979), p. 17; Solution to Problem 410, ibid., Vol. 5, No. 10 (1979), pp. 296-299.

Cf. A000079, A007377, A020862, A130694, A130696.

a:= proc(n) option remember; local k; for k from a(n-1)
      while min((p-> seq(coeff(p, x, j), j=0..9))(add(
            x^i, i=convert(2^k, base, 10))))Alois P. Heinz, Apr 22 2022

s = Table[Min[DigitCount[2^n, 10, Range[0, 9]]], {n, 1, 2500}]; Table[FirstPosition[s, _?(# >= n &)], {n, 1, Max[s]}] // Flatten

from sympy import ceiling, log
def A352040(n):
    k = 10*n-1+int(ceiling((10*n-1)*log(5,2)))
    s = str(c := 2**k)
    while any(s.count(d) < n for d in '0123456789'):
        c *= 2
        k += 1
        s = str(c)
    return k   # Chai Wah Wu, Apr 16 2022

Conjecture: a(n) ~ c*n, where c = 10*log_2(10) = 33.21928... .

a(n) >= (10n-1)*log_2(10), i.e., c = 10*log_2(10) is a lower bound on the asymptotic growth rate. - Chai Wah Wu, Apr 16 2022

A014257 Product of digits of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 6, 6, 24, 16, 60, 10, 0, 0, 0, 144, 576, 2016, 2700, 0, 384, 5120, 0, 0, 0, 0, 24696, 21600, 0, 18816, 691200, 0, 0, 1032192, 11757312, 27993600, 6096384, 39191040, 64012032, 15240960, 0, 387072000, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125971200, 0, 4180377600, 0, 0, 0

Offset: 0

N. J. A. Sloane

Cf. A000079, A007377.

Array[ Times @@ IntegerDigits[ 2^# ]&, 100 ]

a(n) = vecprod(digits(2^n)); \\ Michel Marcus, Nov 18 2019

More terms from Michel Marcus, Nov 18 2019

A117060 Mersenne numbers for which the product of the digits is not zero.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 8191, 16383, 32767, 65535, 262143, 524287, 16777215, 33554431, 134217727, 268435455, 2147483647, 4294967295, 8589934591, 17179869183, 34359738367, 68719476735, 137438953471, 549755813887, 562949953421311

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 16 2006

These are all the Mersenne numbers for which no digit is 0. The indices of these numbers are A007377.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..35

Cf. A000225, A007377.

Flatten[Table[If[First@Union@IntegerDigits[2^n - 1] != 0, 2^n - 1, {}], {n, 100}]] (* Arkadiusz Wesolowski, Sep 04 2011 *)

for(n=1, 99, if(vecmin(eval(Vec(Str(2^n)))), print1(2^n-1", "))) \\ Charles R Greathouse IV, Jun 30 2011

use bignum;
for(1..99) {
  if((1<<$_) =~ /^[1-9]+$/) {
    print(((1 << $_) - 1) . ", ")
  }
} # Charles R Greathouse IV, Jun 30 2011

a(n) = 2^(A007377(n)) - 1. [Arkadiusz Wesolowski, Jun 28 2011]

Offset corrected by Arkadiusz Wesolowski, Jun 28 2011

Corrected by Arkadiusz Wesolowski, Oct 03 2011

A131614 Numbers k such that the decimal expansion of 3^k contains no 8.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 16, 17, 19, 21, 33, 36, 51, 55, 56, 100

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 100 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377.

[n: n in [0..1000] | not 8 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 8] &]]

Adapted Mma and initial 0 added by Vincenzo Librandi, May 06 2015

A131615 Numbers k such that the decimal expansion of 3^k contains no 7.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 12, 13, 17, 21, 22, 24, 26, 30, 32, 33, 36, 42, 46, 66, 101

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 101 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377.

[n: n in [0..1000] | not 7 in Intseq(3^n)]; // Vincenzo Librandi, May 06 2015

Select[ Range@10000, FreeQ[ IntegerDigits[3^# ],7] &]

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

A131616 Numbers k such that the decimal expansion of 3^k contains no 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 15, 18, 23, 32, 35, 42, 55, 104

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 104 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377.

[n: n in [0..1000] | not 6 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 6] &]]

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

A131617 Numbers k such that the decimal expansion of 3^k contains no 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 14, 15, 16, 17, 18, 19, 20, 23, 25, 29, 31, 41, 44, 52, 58, 81, 91

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 91 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377.

[n: n in [0..1000] | not 5 in Intseq(3^n) ]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 5] &]]

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

A131618 Numbers k such that the decimal expansion of 3^k contains no 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 22, 33, 34, 38, 46, 49, 75, 106

Offset: 1

Shyam Sunder Gupta, Sep 01 2007

I conjecture that 106 is the last term.

Cf. similar sequences listed in A131613.

Cf. A007377.

[n: n in [0..1000] | not 4 in Intseq(3^n)]; // Vincenzo Librandi, May 06 2015

Join[{0}, Select[ Range@10000, FreeQ[ IntegerDigits[3^# ], 4] &]]

Initial 0 added and Mathematica code adapted by Vincenzo Librandi, May 06 2015

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A240963 Numbers n such that the decimal expansion of n^n contains no zeros.

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A298607 Powers of 2 with the digit '0' in their decimal expansion.

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A352040 a(n) is the least number k such that 2^k contains each of the 10 digits at least n times.

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A014257 Product of digits of 2^n.

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A117060 Mersenne numbers for which the product of the digits is not zero.

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A131614 Numbers k such that the decimal expansion of 3^k contains no 8.

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Magma

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A131615 Numbers k such that the decimal expansion of 3^k contains no 7.

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Magma

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A131616 Numbers k such that the decimal expansion of 3^k contains no 6.

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