A007931 -id:A007931 - cp's OEIS frontend

A235860 Minimal representation (considered minimal in any canonical base b >= 3) of n in a binary system using two distinct digits "1" and "2", not allowing zeros, where a digit d in position p (p = 1,2,3,...,n) represents the value d^p.

1, 2, 12, 112, 21, 22, 122, 1122, 11122, 211, 212, 1212, 221, 222, 1222, 11222, 111222, 1111222, 2111, 2112, 12112, 2121, 2122, 12122, 112122, 2211, 2212, 12212, 2221, 2222, 12222, 112222, 1112222, 11112222, 111112222, 21111, 21112, 121112, 21121, 21122

Offset: 1

Robin Garcia, Jan 16 2014

			a(4) = 112 because 1^3 + 1^2 + 2^1 = 4.
36(10) in base 10 is represented as 21111 in this base because 2^5 + 1^4 + 1^3 + 1^2 + 1^1 = 36. It could also be represented as 1111112222. The minimal representation, considered in base 10, is chosen.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A007931, A236547, A237662, A237816, A237454.

t = Range[1000]*0; Do[d=1+IntegerDigits[k, 2, n]; dd = FromDigits@d; v = Total[ Reverse[d]^ Range[n]]; If[0 < v <= 1000 && (t[[v]] == 0 || dd < t[[v]]), t[[v]] = dd], {n,17}, {k, 0, 2^n-1}]; t (* first 1000 terms, Giovanni Resta, Jan 16 2014 *)

A239019 Numbers which are not primitive words over the alphabet {0,...,9} (when written in base 10).

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4444, 4545, 4646, 4747, 4848, 4949

Offset: 1

M. F. Hasler, Mar 08 2014

A word is primitive iff it is not a power, i.e., repetition, of a subword. The only non-primitive words with a prime number of letters (here: digits) are the repdigit numbers. Thus, the first nontrivial terms of this sequence are 1010,1212,...
This sequence does *not* contain all non-primitive words over the alphabet {0,...,9}, namely, it excludes those which would be numbers with leading zeros: 00,000,0000,0101,0202,...
Lists of non-primitive words over a sub-alphabet of {1...9}, like A213972, A213973, A213974, A239018, ... are given as intersection of this with the set of all words in that alphabet, e.g., A007931, A032810, A032917, A007932, ...

Robert Israel, Table of n, a(n) for n = 1..10000

F:= proc(d) local p,R,q;
  R:= {seq(x*(10^d-1)/9, x=1..9)};
  for p in numtheory:-factorset(d) minus {d} do
    q:= d/p;
    R:= R union {seq(x*(10^d-1)/(10^q-1),x=10^(q-1)..10^q-1)};
  od:
  sort(convert(R,list))
end proc:
[seq(op(F(i)),i=2..4)]; # Robert Israel, Nov 14 2017

is_A239019(n)=fordiv(#n=digits(n),L,L<#n && n==concat(Col(vector(#n/L,i,1)~*vecextract(n,2^L-1))~)&&return(1))

A284921 Numbers with digits 2 and 7 only.

Original entry on oeis.org

2, 7, 22, 27, 72, 77, 222, 227, 272, 277, 722, 727, 772, 777, 2222, 2227, 2272, 2277, 2722, 2727, 2772, 2777, 7222, 7227, 7272, 7277, 7722, 7727, 7772, 7777, 22222, 22227, 22272, 22277, 22722, 22727, 22772, 22777, 27222, 27227, 27272, 27277, 27722, 27727

Offset: 1

Jaroslav Krizek, Apr 05 2017

Prime terms are in A020459.

Cf. Numbers with digits 2 and k only for k = 0 - 1 and 3 - 9: A169965 (k = 0), A007931 (k = 1), A032810 (k = 3), A284920 (k = 4), A072961 (k = 5), A284632 (k = 6), this sequence (k = 7), A284922 (k = 8), A284923 (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {2, 7}]

Flatten@ Array[FromDigits /@ Tuples[{2, 7}, #] &, 5] (* Michael De Vlieger, Apr 06 2017 *)

A346303 Positions of words in A076478 that start with 0 and end with 0.

Original entry on oeis.org

1, 3, 7, 9, 15, 17, 19, 21, 31, 33, 35, 37, 39, 41, 43, 45, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179

Offset: 1

Clark Kimberling, Jul 21 2021

The sequences A346303, A171757, A346304, A346305 partition the positive integers. See A076478 for a guide to related sequences.

The position of the words 0, 00, 000, 0000, ... (all-zero) in A076478 is given by A000225. So this sequence is given by the subsequence A000225 and filling the gaps halfway up (until the leading 0 toggles) with odd numbers, from 7 to 9, from 15 to 21, from 31 to 45 etc. So the first differences are 2, 4, 2, 6, 2, 2, 2, 10, 2, 2, 2, 2, 2, 2, 2, 18, 2, 2, ... which is A052548 with 1, 3, 7, 15,... (A000225) intermediate 2's. - R. J. Mathar, Sep 08 2021

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(3) = 7.

Cf. A007931, A076478, A171757, A346304, A346305.

Mathematica
```
(See A007931.)
```

A346304 Positions of words in A076478 that start with 1 and end with 0.

Original entry on oeis.org

5, 11, 13, 23, 25, 27, 29, 47, 49, 51, 53, 55, 57, 59, 61, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237

Offset: 1

Clark Kimberling, Jul 21 2021

The sequences A346303, A171757, A346304, A346305 partition the positive integers. See A076478 for a guide to related sequences.

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 5.

Cf. A007931, A076478, A171757, A346303, A346305.

Mathematica
```
(See A007931.)
```

A059943 Toss a fair coin and calculate the expected time until the n-th possible finite sequence of Heads and Tails first appears (ordered by length of sequence and alphabetical order so H, T, HH, HT, TH, TT, HHH, etc.).

Original entry on oeis.org

2, 2, 6, 4, 4, 6, 14, 8, 10, 8, 8, 10, 8, 14, 30, 16, 18, 16, 18, 20, 18, 16, 16, 18, 20, 18, 16, 18, 16, 30, 62, 32, 34, 32, 38, 32, 34, 32, 34, 36, 42, 32, 34, 36, 34, 32, 32, 34, 36, 34, 32, 42, 36, 34, 32, 34, 32, 38, 32, 34, 32, 62, 126, 64, 66, 64, 70, 64, 66, 64, 70, 72

Offset: 1

Henry Bottomley, Feb 14 2001

Note the apparent paradox: HHHHHH, HTHTHT and HHHFFF are all equally likely to appear in six tosses of the coin (1/64) and in a long sequence each is expected to appear as a subsequence roughly as many times as the others, but the expected time for HHHHHH to first appear (126) is almost twice as long as for HHHFFF (64), with HTHTHT between the two (84). This is related to the fact that in a sequence of, say, 8 successive tosses, HHHHHH could appear as a subsequence 3 times simultaneously, HTHTHT twice but HHHTTT only once.

			a(35)=38 since the expected time from xxxHHTH to completion of xxxHHTHH is 20, from xxxHHT to completion is 30, from xxxHH to completion is 32, from xxxH to completion is 36 and from xxx to completion is 38 (xxx is an earlier subsequence, perhaps empty, which cannot contribute to completion).

M. Gardner, Chapter 5 in Time Travel and Other Mathematical Bewilderments, W. H. Freeman, 1988, pp. 63-67.
Michael Hochster in sci.math and sci.stat.math quoting from Stochastic Processes by Sheldon Ross.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007931, A059941, A059942.

a059943 = (* 2) . a059942  -- Reinhard Zumkeller, Apr 03 2014

a[n_] := (id = Drop[ IntegerDigits[n+1, 2], 1]+1; an={}; Do[PrependTo[an, If[Take[id, k] == Take[id, -k], 1, 0]], {k, 1, Length[id]}]; 2*FromDigits[an, 2]); Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Nov 21 2011 *)

a(n) = 2*A059942(n). For the n-th sequence S (e.g., the 35th is HHTHH), create the set X consisting of subsequences of S which appear both at the beginning and end of S (e.g., X={H, HH, HHTHH}), then a(n) = sum_x(2^length(x)|x is in X) (e.g., a(35)=2^1+2^2+2^5=38).

A213084 Numbers consisting of ones and eights.

Original entry on oeis.org

1, 8, 11, 18, 81, 88, 111, 118, 181, 188, 811, 818, 881, 888, 1111, 1118, 1181, 1188, 1811, 1818, 1881, 1888, 8111, 8118, 8181, 8188, 8811, 8818, 8881, 8888, 11111, 11118, 11181, 11188, 11811, 11818, 11881, 11888, 18111, 18118, 18181, 18188, 18811, 18818

Offset: 1

Jens Ahlström, Jun 05 2012

One and eight begin with vowels. The subsequence of primes begins 11, 181, 811, 1181, 1811, 8111. - Jonathan Vos Post, Jun 14 2012

Alois P. Heinz, Table of n, a(n) for n = 1..16382

Cf. A020456 (primes in this sequence).

Cf. numbers consisting of 1s and ks: A007088 (k=0), A007931 (k=2), A032917 (k=3), A032822 (k=4), A276037 (k=5), A284293 (k=6), A276039 (k=7), A284294 (k=9).

Flatten[Table[FromDigits/@Tuples[{1,8},n],{n,5}]] (* Harvey P. Dale, Aug 27 2014 *)

is(n) = #setintersect(vecsort(digits(n), , 8), [0, 2, 3, 4, 5, 6, 7, 9])==0 \\ Felix Fröhlich, Sep 09 2019

res = []
i = 0
while len (res) < 260:
    for c in str(i):
        if c in '18':
            continue
        else:
            break
    else:
        res.append(i)
    i = i + 1
print(res)

def a(n): return int(bin(n+1)[3:].replace('1', '8').replace('0', '1'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Jun 26 2025

A284294 Numbers using only digits 1 and 9.

Original entry on oeis.org

1, 9, 11, 19, 91, 99, 111, 119, 191, 199, 911, 919, 991, 999, 1111, 1119, 1191, 1199, 1911, 1919, 1991, 1999, 9111, 9119, 9191, 9199, 9911, 9919, 9991, 9999, 11111, 11119, 11191, 11199, 11911, 11919, 11991, 11999, 19111, 19119, 19191, 19199, 19911, 19919

Offset: 1

Jaroslav Krizek, Mar 25 2017

Product of digits of terms is a power of 9; subsequence of A284295.

Prime terms are in A020457.

Alois P. Heinz, Table of n, a(n) for n = 1..16382

Cf. Numbers using only digits 1 and k for k = 0 and k = 2 - 9: A007088 (k = 0), A007931 (k = 2), A032917 (k = 3), A032822 (k = 4) , A276037 (k = 5), A284293 (k = 6), A276039 (k = 7), A213084 (k = 8), this sequence (k = 9).

[n: n in [1..20000] | Set(IntegerToSequence(n, 10)) subset {1, 9}];

Join @@ (FromDigits /@ Tuples[{1,9}, #] & /@ Range[5]) (* Giovanni Resta, Mar 25 2017 *)

The sum of first 2^n terms is (5*20^n + 38*10^n - 95*2^n + 1420)/171. - Giovanni Resta, Mar 25 2017

A291073 Take n-th string over {1,2} in lexicographic order and apply the Watanabe tag system {00, 1110} described in A291068 (but adapted to the alphabet {1,2}) just once.

Original entry on oeis.org

-1, 21, 1, 1, 221, 221, 11, 11, 11, 11, 2221, 2221, 2221, 2221, 111, 211, 111, 211, 111, 211, 111, 211, 12221, 22221, 12221, 22221, 12221, 22221, 12221, 22221, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111

Offset: 1

N. J. A. Sloane, Aug 18 2017

sign

Cf. A007931, A284116, A291067, A291068, A291069.

Cf. also A289673, A291072, A291074.

Maple
```
See A291072.
```

A291074 Take n-th string over {1,2} in lexicographic order and apply the Watanabe tag system {00, 0111} described in A291069 (but adapted to the alphabet {1,2}) just once.

Original entry on oeis.org

-1, 22, 1, 1, 222, 222, 11, 11, 11, 11, 1222, 1222, 1222, 1222, 111, 211, 111, 211, 111, 211, 111, 211, 11222, 21222, 11222, 21222, 11222, 21222, 11222, 21222, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111

Offset: 1

N. J. A. Sloane, Aug 18 2017

sign

Cf. A007931, A284116, A291067, A291068, A291069.

Cf. also A289673, A291072, A291073.

Maple
```
See A291072.
```

cp's OEIS Frontend

A235860 Minimal representation (considered minimal in any canonical base b >= 3) of n in a binary system using two distinct digits "1" and "2", not allowing zeros, where a digit d in position p (p = 1,2,3,...,n) represents the value d^p.

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A239019 Numbers which are not primitive words over the alphabet {0,...,9} (when written in base 10).

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Maple

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A284921 Numbers with digits 2 and 7 only.

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Magma

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A346303 Positions of words in A076478 that start with 0 and end with 0.

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Mathematica

A346304 Positions of words in A076478 that start with 1 and end with 0.

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Mathematica

A059943 Toss a fair coin and calculate the expected time until the n-th possible finite sequence of Heads and Tails first appears (ordered by length of sequence and alphabetical order so H, T, HH, HT, TH, TT, HHH, etc.).

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A213084 Numbers consisting of ones and eights.

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Python

A284294 Numbers using only digits 1 and 9.

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Magma