A032810 -id:A032810 - cp's OEIS frontend

A213974 List of imprimitive words over the alphabet {2,3}.

22, 33, 222, 333, 2222, 2323, 3232, 3333, 22222, 33333, 222222, 223223, 232232, 232323, 233233, 322322, 323232, 323323, 332332, 333333, 2222222, 3333333, 22222222, 22232223, 22322232, 22332233, 23222322, 23232323, 23322332, 23332333, 32223222, 32233223, 32323232, 32333233, 33223322, 33233323, 33323332, 33333333

Offset: 1

N. J. A. Sloane, Jun 30 2012

nonn

A word w is primitive if it cannot be written as u^k with k>1; otherwise it is imprimitive.
The {0,1} version of this sequence is 00, 11, 000, 111, 0000, 0101, 1010, 1111, 00000, 11111, 000000, 001001, 010010, 010101, 011011, 100100, 101010, 101101, 110110, 111111 but this cannot be included as a sequence in the OEIS since it contains nonzero "numbers" beginning with 0.
This sequence results from A213973 by replacing each digit 1 by 2, and from A213972 by replacing all digits 2 by 3 and all digits 1 by 2. - M. F. Hasler, Mar 10 2014

A. de Luca and S. Varricchio, Finiteness and Regularity in Semigroups and Formal Languages, Monographs in Theoretical Computer Science, Springer-Verlag, Berlin, 1999. See p. 10.

Cf. A213969, A213970, A213971, A213972, A213973.

for(n=1, 8, p=vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [2, 3]), is_A239017(m=d*p)||print1(m", "))) \\ M. F. Hasler, Mar 10 2014

Equals A032810 intersect A239018. - M. F. Hasler, Mar 10 2014

More terms from M. F. Hasler, Mar 10 2014

A143967 Numbers containing only digits 3 or 7 in decimal representation.

Original entry on oeis.org

3, 7, 33, 37, 73, 77, 333, 337, 373, 377, 733, 737, 773, 777, 3333, 3337, 3373, 3377, 3733, 3737, 3773, 3777, 7333, 7337, 7373, 7377, 7733, 7737, 7773, 7777, 33333, 33337, 33373, 33377, 33733, 33737, 33773, 33777, 37333, 37337, 37373, 37377

Offset: 1

Reinhard Zumkeller, Sep 06 2008

See A020463 for primes.

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

Cf. A004086, A020463, A032810.

a143967 = f 0 . (+ 1) where
   f y 1 = a004086 y
   f y x = f (10 * y + 3 + 4 * r) x' where (x', r) = divMod x 2
-- Reinhard Zumkeller, Mar 18 2015

Table[FromDigits/@Tuples[{3,7},n],{n,5}]//Flatten (* Harvey P. Dale, Aug 28 2017 *)

a(n) = f(n+1, 0) with f(n, x) = if n=1 then A004086(x) else f(floor(n/2), 10*x + 3 + 4*(n mod 2)).

A248907 Numbers consisting only of digits 2 and 3, ordered according to the value obtained when the digits are interspersed with (right-associative) ^ operators.

Original entry on oeis.org

2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332, 2222, 3222, 233, 333, 2322, 3322, 2223, 3223, 2232, 3232, 2323, 3323, 2332, 3332, 22222, 32222, 23222, 33222, 2233, 3233, 2333, 3333, 22322, 32322, 23322, 33322, 22223, 32223, 23223, 33223, 22232, 32232

Offset: 1

Vladimir Reshetnikov and Reinhard Zumkeller, Mar 18 2015

A256179(n) is found by treating the digits of a(n) as power towers. So for example, a(11) = 323, so A256179(11) = 6561 because 3^(2^3) = 6561. - Bob Selcoe, Mar 18 2015

This is a permutation of the list A032810 (numbers having only digits 2 and 3) in the sense that is a list with exactly the same terms but in different order, namely such that the ("power tower") function A256229 yields an increasing sequence. The permutation of the indices is given by A185969, cf. formula. - M. F. Hasler, Mar 21 2015

Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015.
Vladimir Reshetnikov et al., Power towers of 2 and 3 - looking for a proof, on StackExchange.com, Mar 19 2015

Cf. A032810, A185969, A256179, A256229.

For another version, see A299229 (each digit is a separate term).

Haskell
```
a248907 = a032810 . a185969
```

ClearAll[a, p];
p[d_, n_] := d 10^IntegerLength[n] + n;
a[n_ /; n <= 12] := a[n] = {2, 3, 22, 23, 32, 222, 33, 322, 223, 232, 323, 332}[[n]];
a[n_ /; OddQ[n]]  := a[n] = p[2, a[(n - 1)/2]];
a[n_] := a[n] = p[3, a[(n - 2)/2]];
Array[a, 100]

vecsort(A032810,(a,b)->A256229(a)>A256229(b)) \\ Assuming that A032810 is defined as a vector. Append [1..N] if the vector A032810 has too many (thus too large) elements: recall that 33333 => 3^(3^(3^(3^3))). - M. F. Hasler, Mar 21 2015

a(n) = A032810(A185969(n)).

Edited by M. F. Hasler, Mar 21 2015

A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

Original entry on oeis.org

1, 1, 11, 11, 11, 11, 111, 111, 111, 111, 111, 111, 111, 111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111

Offset: 1

M. F. Hasler, Mar 21 2015

nonn

Yields the length of the n-th (nonempty) binary word (or word over any 2-letter alphabet, like A007931 or A032810 or A032834) in tally mark notation (A000042).

Felix Fröhlich, Table of n, a(n) for n = 1..10000

lim = 5; lst = Table[(10^n - 1)/9, {n, 0, lim}]; Reap@ For[i = 1, i <= lim, i++, Sow@ Table[lst[[i + 1]], {d, 2^i}]] // Flatten // Rest (* Michael De Vlieger, Apr 01 2015 *)

PARI
```
a(n)=10^#binary(n+1)\90
```

def A256077(n): return (10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Nov 07 2024

a(n) = A002275(A000523(n+1)) = A032810(n)-A007931(n) = A032834(n)-A032810(n), etc.

A284634 Numbers with digits 4 and 6 only.

Original entry on oeis.org

4, 6, 44, 46, 64, 66, 444, 446, 464, 466, 644, 646, 664, 666, 4444, 4446, 4464, 4466, 4644, 4646, 4664, 4666, 6444, 6446, 6464, 6466, 6644, 6646, 6664, 6666, 44444, 44446, 44464, 44466, 44644, 44646, 44664, 44666, 46444, 46446, 46464, 46466, 46644, 46646

Offset: 1

Jaroslav Krizek, Apr 02 2017

All terms are even.

Numbers n with digits 6 and k only for k = 0 - 5 and 7 - 9: A204093 (k = 0), A284293 (k = 1), A284632 (k = 2), A284633 (k = 3), this sequence (k = 4), A256291 (k = 5), A256292 (k = 7), A284635 (k = 8), A284636 (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {4, 6}]

Table[FromDigits /@ Tuples[{4, 6}, n], {n, 5}] // Flatten (* or *)
Select[Range@ 50000, Total@ Pick[DigitCount@ #, {0, 0, 0, 1, 0, 1, 0, 0, 0, 0}, 0] == 0 &] (* Michael De Vlieger, Apr 02 2017 *)

a(n) = 2 * A032810(n).

A284920 Numbers with digits 2 and 4 only.

Original entry on oeis.org

2, 4, 22, 24, 42, 44, 222, 224, 242, 244, 422, 424, 442, 444, 2222, 2224, 2242, 2244, 2422, 2424, 2442, 2444, 4222, 4224, 4242, 4244, 4422, 4424, 4442, 4444, 22222, 22224, 22242, 22244, 22422, 22424, 22442, 22444, 24222, 24224, 24242, 24244, 24422, 24424

Offset: 1

Jaroslav Krizek, Apr 05 2017

All terms are even.

Cf. Numbers with digits 2 and k only for k = 0 - 1 and 3 - 9: A169965 (k = 0), A007931 (k = 1), A032810 (k = 3), this sequence (k = 4), A072961 (k = 5), A284632 (k = 6), A284921 (k = 7), A284922 (k = 8), A284923 (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {2, 4}]

Flatten@ Array[FromDigits /@ Tuples[{2, 4}, #] &, 5] (* Michael De Vlieger, Apr 06 2017 *)

a(n) = 2 * A007931(n).

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))

A256179 Sequence of power towers in ascending order, using all possible permutations of 2's and 3's.

Original entry on oeis.org

2, 3, 4, 8, 9, 16, 27, 81, 256, 512, 6561, 19683, 65536, 43046721, 134217728, 7625597484987, 2417851639229258349412352, 443426488243037769948249630619149892803, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset: 1

Bob Selcoe, Mar 18 2015

nonn

a(n) is found by treating the digits of A248907(n) as power towers, so the sequence starts 2, 3, 2^2=4, 2^3=8, 3^2=9, 2^(2^2)=16, 3^3=27, 3^(2^2)=81, 2^(2^3)=256...

			a(12) = 19683 because A248907(12) = 332, and 3^(3^2) = 19683.
a(23) = 2^3^2^3 = 11423...73952 (1976 digits), because A248907(23) = 2323.

Pontus von Brömssen, Table of n, a(n) for n = 1..22
Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015.

Cf. A032810, A075877, A185969, A248907, A256229.

A256179(n)=A256229(A248907[n]) \\ where A248907 is assumed to be defined as vector. - M. F. Hasler, Mar 19 2015

A256179 = A256229 o A248907 = A256229 o A032810 o A185969, i.e., a(n) = A256229(A248907(n)) = A256229(A032810(A185969(n))).

Recurrence: a(1)=2, a(2)=3, a(3)=2^2, a(4)=2^3, a(5)=3^2, a(6)=2^(2^2), a(7)=3^3, a(8)=3^(2^2), a(9)=2^(2^3), a(10)=2^(3^2), a(11)=3^(2^3), a(12)=3^(3^2); and for n>6, a(2n)=3^a(n-1), a(2n-1)=2^a(n-1). - Vladimir Reshetnikov, Mar 19 2015

More terms from M. F. Hasler, Mar 19 2015

A284636 Numbers with digits 6 and 9 only.

Original entry on oeis.org

6, 9, 66, 69, 96, 99, 666, 669, 696, 699, 966, 969, 996, 999, 6666, 6669, 6696, 6699, 6966, 6969, 6996, 6999, 9666, 9669, 9696, 9699, 9966, 9969, 9996, 9999, 66666, 66669, 66696, 66699, 66966, 66969, 66996, 66999, 69666, 69669, 69696, 69699, 69966, 69969

Offset: 1

Jaroslav Krizek, Apr 02 2017

All terms are composite.

All terms are divisible by 3. - Michael S. Branicky, Jun 09 2021

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A032810.

Numbers n with digits 6 and k only for k = 0 - 5 and 7 - 9: A204093 (k = 0), A284293 (k = 1), A284632 (k = 2), A284633 (k = 3), A284634 (k = 4), A256291 (k = 5), A256292 (k = 7), A284635 (k = 8), this sequence (k = 9).

[n: n in [1..100000] | Set(IntegerToSequence(n, 10)) subset {6, 9}]

Table[FromDigits /@ Tuples[{6, 9}, n], {n, 5}] // Flatten (* or *)
Select[Range@ 70000, Total@ Pick[DigitCount@ #, {0, 0, 0, 0, 0, 1, 0, 0, 1, 0}, 0] == 0 &] (* Michael De Vlieger, Apr 02 2017 *)

a(n) = {
  my(z, e = logint(n+1,2,&z),
     t1 = 9 * subst(Pol(binary(n+1-z),'x), 'x, 10),
     t2 = 6 * subst(Pol(binary(2*z-2-n),'x), 'x, 10));
  t1+t2;
};
vector(44, n, a(n)) \\ Gheorghe Coserea, Apr 04 2017

def a(n): return int(bin(n+1)[3:].replace('0', '6').replace('1', '9'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Jun 09 2021

a(n) = 3 * A032810(n).

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 *)

cp's OEIS Frontend

A213974 List of imprimitive words over the alphabet {2,3}.

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A143967 Numbers containing only digits 3 or 7 in decimal representation.

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A248907 Numbers consisting only of digits 2 and 3, ordered according to the value obtained when the digits are interspersed with (right-associative) ^ operators.

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A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

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A284634 Numbers with digits 4 and 6 only.

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Magma

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A284920 Numbers with digits 2 and 4 only.

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Magma

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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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A256179 Sequence of power towers in ascending order, using all possible permutations of 2's and 3's.

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