A111066 -id:A111066 - cp's OEIS frontend

A086066 a(n) = Sum_{d in D(n)} 2^d, where D(n) = set of digits of n in decimal representation.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 3, 2, 6, 10, 18, 34, 66, 130, 258, 514, 5, 6, 4, 12, 20, 36, 68, 132, 260, 516, 9, 10, 12, 8, 24, 40, 72, 136, 264, 520, 17, 18, 20, 24, 16, 48, 80, 144, 272, 528, 33, 34, 36, 40, 48, 32, 96, 160, 288, 544, 65, 66, 68, 72, 80

Offset: 0

Reinhard Zumkeller, Jul 08 2003

For bitwise logical operations AND and OR:
a(m) = (a(m) AND a(n)) iff D(m) is a subset of D(n),
(a(m) AND a(n)) = 0 iff D(m) and D(n) are disjoint,
a(m) = (a(m) OR a(n)) iff D(n) is a subset of D(m),
a(m) = a(n) iff D(m) = D(n);
A086067(n) = A007088(a(n)).
From Reinhard Zumkeller, Sep 18 2009: (Start)
a(A052382(n)) mod 2 = 0; a(A011540(n)) mod 2 = 1;
for n > 0: a(A000004(n))=1, a(A000042(n))=2, a(A011557(n))=3, a(A002276(n))=4, a(A111066(n))=6, a(A002277(n))=8, a(A002278(n))=16, a(A002279(n))=32, a(A002280(n))=64, a(A002281(n))=128, a(A002282(n))=256, a(A002283(n))=512;
a(n) <= 1023. (End)

			n=242, D(242) = {2,4}: a(242) = 2^2 + 2^4 = 20.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

A086066 := proc(n) local d: if(n=0)then return 1: fi: d:=convert(convert(n,base,10),set): return add(2^d[j],j=1..nops(d)): end: seq(A086066(n),n=0..64); # Nathaniel Johnston, May 31 2011

A214218 List of words over {1,2} with equal numbers of 1's and 2's.

Original entry on oeis.org

12, 21, 1122, 1212, 1221, 2112, 2121, 2211, 111222, 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111, 11112222, 11121222, 11122122, 11122212, 11122221

Offset: 1

N. J. A. Sloane, Jul 18 2012

Of course the empty word also has this property.
All of these, interpreted as decimal integers are divisible by 3, because each pair of "1" and "2" contributes a digital sum of 3, hence the total is divisible by 3. Is there a semiprime in the sequence after 21? - Jonathan Vos Post, Jul 18 2012
The semiprime subsequence contains 21, 11222121, 12122211, 21221121, 22211121, 22212111, and continues with 14 10-digit entries etc. - R. J. Mathar, Jul 19 2012

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A007931, A111066.

sort([seq(seq((10^(2*d)-1)/9+add(10^i,i=s),s=combinat:-choose([$0..(2*d-1)],d)),d=1..4)]); # Robert Israel, Jan 02 2018

Sort[FromDigits/@Flatten[Table[Permutations[PadRight[{},2n,{1,2}]],{n,3}],1]] (* Harvey P. Dale, Aug 30 2016 *)

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations as mp
def agen():
    for d in count(2, 2):
        for s in mp("1"*(d//2) + "2"*(d//2), d):
            yield int("".join(s))
print(list(islice(agen(), 33))) # Michael S. Branicky, Dec 21 2021

A318700 Positive numbers that contain odd and even digits.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 114, 116, 118, 120, 121, 122

Offset: 1

Enrique Navarrete, Aug 31 2018

The sequence of first differences takes on the values {1, 2, 3} only, and each of these values occurs infinitely often (the values 1 and 2 are clear; for the value 3, note that consecutive numbers such as 199..9, 200..0 and 399..9, 400..0 that are excluded from the sequence occur infinitely often).
Numbers n such that A065031(n) is a term of A111066. - Felix Fröhlich, Sep 01 2018
Nonnegative integers excluding those such that digits in their decimal representation share all the same parity. - R. J. Cano, Sep 10 2018

			49 and 50 are in the sequence but 19 and 20 are not.

R. J. Cano, PARI program

Cf. A065031, A111066, A030141, A054684, A227870.

Select[Range@ 122, Union[Mod[IntegerDigits[#], 2]] == {0, 1} &] (* Michael De Vlieger, Sep 04 2018 *)

is(n) = my(d=digits(n), v=[]); if(n < 10, return(0)); for(k=1, #d, v=concat(v, [d[k]%2])); vecmin(v)!=vecmax(v) \\ Felix Fröhlich, Sep 01 2018

PARI
```
See Cano link.
```

A380435 Erase digit 0 from decimal expansion of n. Then repeatedly apply the number of divisor function (A000005) onto each digit until a stationary value is reached. a(n) is the final stationary value (if it is reached for all digits).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 11, 12, 12, 12, 12, 12, 12, 12, 12, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22

Offset: 1

Ctibor O. Zizka, Jan 24 2025

The number of iterations is 0, 1, 2, 3 for numbers containing the highest digits (1, 2), (3,5,7), (4, 9), (6, 8). n >= a(n) >= 1.

			For n = 21 a(21) = 21.
For n = 408 we iterate 48 --> 34 --> 23 --> 22, thus, after 3 iterations, a(408) = 22.

Cf. A002275, A002276 - A002283, A004719, A007931, A111066.

a[n_] := FromDigits[IntegerDigits[n] /. {0 -> Nothing, ?(# > 1 &) -> 2}]; Array[a, 100] (* _Amiram Eldar, Jan 24 2025 *)

a(A007931(n)) = A007931(n).
For r = 1, k >= 0:
a(10^k) = 1
a((10^k - 1)/9) = (10^k - 1)/9.
For r from [2, 9], k >= 0:
a(r*10^k) = 2.
a(r*(10^k - 1)/9) = 2*(10^k - 1)/9.

cp's OEIS Frontend

A086066 a(n) = Sum_{d in D(n)} 2^d, where D(n) = set of digits of n in decimal representation.

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A214218 List of words over {1,2} with equal numbers of 1's and 2's.

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A318700 Positive numbers that contain odd and even digits.

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Mathematica

PARI

PARI

A380435 Erase digit 0 from decimal expansion of n. Then repeatedly apply the number of divisor function (A000005) onto each digit until a stationary value is reached. a(n) is the final stationary value (if it is reached for all digits).

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