A130696 -id:A130696 - cp's OEIS frontend

A130694 Exponents of powers of 2 that contain all ten digits.

68, 70, 79, 82, 84, 87, 88, 89, 94, 95, 96, 97, 98, 100, 101, 103, 104, 105, 106, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144

Offset: 1

Greg Dresden, Jul 10 2007

It is believed that every power of 2 beyond 2^86 contains the digit 0.

For k in {51,67,72,76,81,86}, 2^k contains all nonzero digits, but does not contain 0. - Dimiter Skordev, Oct 05 2021

			2^68 = 295147905179352825856.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007377, A000079.

Complement of A130696.

A2 := {}; Do[If[Length[Union[ IntegerDigits[2^ n]]] == 10, A2 = Join[A2, {n}]], {n, 1, 200}]; Print[A2]
Select[Range[200],Min[DigitCount[2^#]]>0&] (* Harvey P. Dale, Aug 03 2019 *)

is_A130694(n)=9<#Set(Vec(Str(1<M. F. Hasler, Aug 25 2012

A043537(A000079(a(n))) = 10. - Reinhard Zumkeller, Jul 29 2007

a(n) = n + 91 for n >= 78 (conjectured). - Chai Wah Wu, Jan 27 2020

Displayed terms double-checked by M. F. Hasler, Aug 25 2012

A272269 Numbers n such that 11^n does not contain all ten decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, 28, 34, 38, 41

Offset: 1

Altug Alkan, Apr 24 2016

Inspiration was the simple form of 11 that is concatenation of 1 and 1. With similar motivation, A130696 focuses on the values of 2^n = (1 + 1)^n. Since this sequence exists in base 10, 11^n*10 is simply concatenation of 11^n and 0. So 11^(n+1) = concat(11^n, 0) + 11^n while 2^(n+1) = 2^n + 2^n.

A030706 is a subsequence. So note that if there is currently no proof of finiteness of A030706, then there is no proof yet of the finiteness of this sequence.

			25 is a term because 11^25 = 108347059433883722041830251 that does not contain digit 6.
26 is not a term because 11^26 = 11^25*10 + 11^25 = 1083470594338837220418302510 + 108347059433883722041830251 = 1191817653772720942460132761 that contains all ten decimal digits.

Cf. A001020, A030706, A130696, A171102.

Select[Range[0, 120], AnyTrue[DigitCount[11^#], # == 0 &] &] (* Michael De Vlieger, Apr 24 2016, Version 10 *)

isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
lista(nn) = for(n=0, nn, if(!isA171102(11^n), print1(n, ", ")));

select( is_A272269(n)=#Set(digits(11^n))<10 ,[0..100]) \\ M. F. Hasler, May 18 2017

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

A236674 Exponents of powers of 3 that do not contain all ten decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 49, 50, 51, 52, 54, 55, 56, 58, 59, 66, 68, 75, 81, 84, 91, 100, 101, 104, 106

Offset: 1

Derek Orr, Jan 29 2014

It is conjectured that 106 is the final number in this sequence. 3^106 contains all digits except for 4.

Complement of A236673.

			3^44 = 984770902183611232881 does not have all ten decimal digits (the 5 is missing), thus 44 is a member of this sequence.

Cf. A130696, A236673.

Select[Range[0, 1000], Union[IntegerDigits[3^#]] != Range[0, 9] &] (* T. D. Noe, Jan 29 2014 *)

def PanDigNot(x):
  a = '1234567890'
  for n in range(10**4):
    count = 0
    for i in a:
      if str(x**n).count(i) > 0:
        count += 1
    if count < len(a):
      print(n)

A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

0, 3, 12, 40, 98, 253, 540, 1199, 2415, 4893, 9268, 17864, 32421, 59265, 104632, 184338, 315414, 540155, 901845, 1504173, 2461932, 4013511, 6443170, 10314675, 16281749, 25608450, 39838855, 61716941, 94682665, 144726102

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

			Illustration of the first few terms
.
0   | 0
3   | 1,  2
12  | 3,  4,  5
40  | 6,  7,  8,  9,  10
98  | 11, 12, 13, 14, 15, 16, 17
253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
.

Cf. A373300, original version, with positive integers A000027.
Cf. A001477, the nonnegative integers.
Cf. A027480, the sequence of row sums for a regular triangle.

Module[{s = -1},
 Table[s +=
   PartitionsP[
    n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
   s (s - 1)/2, {n, 1, 30}]]

A060815 Powers of 2 with each digit a power of 2 (only digits 0,1,2,4,8 are used).

Original entry on oeis.org

2, 4, 8, 128, 1024, 2048

Offset: 1

Amarnath Murthy, Apr 27 2001

Presumably a(6) is the last term. If a(7) exists it is greater than 2^1000000. - Charles R Greathouse IV, Apr 09 2012

			a(3) = 128 = 2^7, each digit of 128 is a power of 2.

Cf. A130696.

Next term > 2^105 - Larry Reeves (larryr(AT)acm.org), May 09 2001

A272322 Numbers n such that n^n does not contain all ten decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 27, 28, 30, 34, 40, 100, 1000, 10000, 100000, 1000000

Offset: 1

Altug Alkan, Apr 25 2016

A240963 is a subsequence.
A011557 is a subsequence. So this sequence is obviously infinite.
Conjecture: the only values > 40 are the powers of 10. - Franklin T. Adams-Watters, Apr 29 2016

			27 is a term because 27^27 = 443426488243037769948249630619149892803 does not contain digit 5.
28 is a term because 28^28 = 33145523113253374862572728253364605812736 does not contain digit 9.

Cf. A000312, A011557, A130696, A171102, A240963, A272269.

{0}~Join~Select[Range[10^4], Times @@ DigitCount[#^#] == 0 &] (* Michael De Vlieger, Apr 25 2016 *)

isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
lista(nn) = for(n=0, nn, if(!isA171102(n^n), print1(n, ", ")));

is(n)=#Set(digits(n^n))<=9 \\ Charles R Greathouse IV, Apr 29 2016

a(31) from Charles R Greathouse IV, Apr 29 2016

a(32) from Charles R Greathouse IV, May 03 2016

cp's OEIS Frontend

A130694 Exponents of powers of 2 that contain all ten digits.

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A272269 Numbers n such that 11^n does not contain all ten decimal digits.

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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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A236674 Exponents of powers of 3 that do not contain all ten decimal digits.

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A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

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Mathematica

A060815 Powers of 2 with each digit a power of 2 (only digits 0,1,2,4,8 are used).

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A272322 Numbers n such that n^n does not contain all ten decimal digits.

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