A005823 -id:A005823 - cp's OEIS frontend

A097260 Numbers whose set of base 14 digits is {0,D}, where D base 14 = 13 base 10.

0, 13, 182, 195, 2548, 2561, 2730, 2743, 35672, 35685, 35854, 35867, 38220, 38233, 38402, 38415, 499408, 499421, 499590, 499603, 501956, 501969, 502138, 502151, 535080, 535093, 535262, 535275, 537628, 537641, 537810, 537823, 6991712

Offset: 1

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 14 for every i.

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

Cf. A001196, A005823, A097251-A097262.

FromDigits[#,14]&/@Tuples[{0,13},6] (* Harvey P. Dale, Sep 22 2011 *)

a(n) = 13*A033050(n).

a(2n) = 14*a(n), a(2n+1) = a(2n)+13.

A190640 Numbers whose base-3 expansion ends in 2 and does not contain any 1's.

Original entry on oeis.org

2, 8, 20, 26, 56, 62, 74, 80, 164, 170, 182, 188, 218, 224, 236, 242, 488, 494, 506, 512, 542, 548, 560, 566, 650, 656, 668, 674, 704, 710, 722, 728, 1460, 1466, 1478, 1484, 1514, 1520, 1532, 1538, 1622, 1628, 1640, 1646, 1676, 1682, 1694, 1700, 1946, 1952, 1964, 1970, 2000, 2006, 2018, 2024, 2108, 2114, 2126

Offset: 1

N. J. A. Sloane, May 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Subsequence of A005823.

Cf. A005836.

Select[Range[2200],Last[IntegerDigits[#,3]]==2&&DigitCount[#,3,1]==0&] (* Harvey P. Dale, Sep 09 2012 *)
FromDigits[#,3]&/@(Join[#,{2}]&/@Tuples[{0,2},7]) (* Harvey P. Dale, Jul 25 2020 *)

is(n)=n%3==2 && setsearch(Set(digits(n,3)), 1)==0 \\ Charles R Greathouse IV, Aug 24 2016

a(n)=2*fromdigits(binary(2*n-1),3) \\ Charles R Greathouse IV, Aug 24 2016

Conjecture: a(n) = A055246(n) + 1. - Michel Marcus, Aug 24 2016

a(n) = A005823(2n). - Charles R Greathouse IV, Aug 24 2016

A249102 Numbers with no 1's in base-7 expansion.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 98, 100, 101, 102, 103, 104, 112, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 128, 129, 130, 131, 132, 133

Offset: 1

Zak Seidov, Oct 21 2014

			14_10 = 20_7, 16_10 = 22_10, 17_10 = 23_7.
14 in base 7 is 20, which contains no 1s, so 14 is in the sequence.
15 in base 7 is 21, which contains one 1, so 15 is not in the sequence.
16 in base 7 is 22, so 16 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Subsequence of A047306. Cf. A023721, A023725, A023729, A023733, A005823. This sequence has no terms in common with A016993.

Select[Range[0, 200], FreeQ[IntegerDigits[#, 7], 1] &] (* Seidov *)
Select[Range[0, 139], DigitCount[#, 7, 1] == 0 &] (* Alonso del Arte, Oct 26 2014 *)

fromdigits(v, b=10)=subst(Pol(v), 'x, b) \\ needed for gp < 2.63 or so
a(n)=a(n)=fromdigits(apply(k->if(k, k+1, 0), digits(n, 6)),7) \\ Charles R Greathouse IV, Oct 30 2014

A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 3, 4, 2, 4, 3, 4, 1, 3, 2, 5, 3, 3, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 1, 2, 3, 3, 2, 4, 5, 6, 3, 4, 3, 6, 4, 4, 5, 6, 2, 3, 5, 6, 4, 6, 5, 6, 3, 6, 5, 6, 4, 6, 5, 6, 1, 4, 2, 4, 3, 5, 3, 7, 2, 4, 4, 4, 5, 5, 6, 7, 3, 5, 4, 7, 3, 5, 6

Offset: 0

Rémy Sigrist, Oct 08 2017

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
The sum of digits of n in base 2^a(n), say s, can be computed without carry in base 2; the Hamming weight of s equals the Hamming weight of n.
a(n) >= A000120(n) for any n > 0.
Apparently, a(n) = A000120(n) iff n = 0 or n belongs to A100290.
a(n) <= A070939(n) for any n >= 0.
For any sequence s of distinct nonnegative integers (s(n) being defined for n >= 0):
- let D_s be defined for any n > 0 by D_s(n) = a(Sum_{k=0..n-1} 2^s(k)),
- then D_s is the discriminator of s as introduced by Arnold, Benkoski, and McCabe in 1985,
- D_s(1) = 1,
- D_s(n) >= n for any n >= 1,
- D_s(n+1) >= D_s(n) for any n >= 1.

			For n=42:
- 42 = 2^5 + 2^3 + 2^1,
- 5 mod 1 = 3 mod 1,
- 5 mod 2 = 3 mod 2,
- 5 mod 3, 3 mod 3 and 1 mod 3 are all distinct,
- hence a(42) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.

Cf. A000041, A000045, A000058, A000069, A000108, A000110, A000120, A000215, A001147, A001566, A001969, A002808, A003095, A005823, A016726, A062383, A070939, A076793, A100290, A133457, A173195, A270097, A270151, A270176, A272633, A272881, A272882, A273037, A273041, A273043, A273044, A273056, A273062, A273064, A273068, A273237, A273377

f:= proc(n) local L,D,k;
  L:= convert(n,base,2);
  L:= select(t -> L[t+1]=1, [$0..nops(L)-1]);
  if nops(L) = 1 then return 1 fi;
  D:= {seq(seq(L[j]-L[i],i=1..j-1),j=2..nops(L))};
  D:= `union`(seq(numtheory:-divisors(i),i=D));
  min({$2..max(D)+1} minus D)
end proc:
0, seq(f(i),i=1..100); # Robert Israel, Oct 08 2017

{0}~Join~Table[Function[r, SelectFirst[Range@ 10, Length@ Union@ Mod[r, #] == Length@ r &]][Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[n, 2]], {n, 86}] (* Michael De Vlieger, Oct 08 2017 *)

a(n) = if (n, my (d=Vecrev(binary(n)), x = []); for (i=1, #d, if (d[i], x = concat(x, i-1))); for (m=1, oo, if (#Set(vector(#x, i, x[i]%m))==#x, return (m))), return (0))

a(2*n) = a(n) for any n >= 0.
a(2^k-1) = k for any k >= 0.
a(n) = 1 iff n = 2^k for some k >= 0.
a(n) = 2 iff n belongs to A173195.
a(Sum_{k=1..n} 2^(k^2)) = A016726(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000069(k)) = A062383(n) for any n >= 1.
a(Sum_{k=0..n} 2^(2^k)) = A270097(n) for any n >= 0.
a(Sum_{k=1..n} 2^A000045(k+1)) = A270151(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000041(k)) = A270176(n) for any n >= 1.
a(A076793(n)) = A272633(n) for any n >= 0.
a(Sum_{k=1..n} 2^A001969(k)) = A272881(n) for any n >= 1.
a(Sum_{k=1..n} 2^A005823(k)) = A272882(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000215(k-1)) = A273037(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000108(k)) = A273041(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001566(k)) = A273043(n) for any n >= 1.
a(Sum_{k=1..n} 2^A003095(k)) = A273044(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000058(k-1)) = A273056(n) for any n >= 1.
a(Sum_{k=1..n} 2^A002808(k)) = A273062(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k!)) = A273064(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k^k)) = A273068(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000110(k)) = A273237(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001147(k)) = A273377(n) for any n >= 1.

A039276 Numbers whose base-3 representation has the same nonzero number of 0's and 2's.

Original entry on oeis.org

6, 11, 15, 19, 21, 32, 34, 38, 42, 46, 48, 56, 58, 60, 64, 66, 72, 89, 95, 97, 101, 103, 105, 113, 115, 119, 123, 127, 129, 137, 139, 141, 145, 147, 153, 167, 169, 173, 175, 177, 181, 183, 191, 193, 195, 199, 201, 207, 217, 219, 225, 260, 266, 268, 278, 284

Offset: 1

Olivier Gérard

Supersequence of A044995.

Cf. A007089, A005823.

Select[Range[300],DigitCount[#,3,0]==DigitCount[#,3,2]>0&] (* Harvey P. Dale, May 10 2011 *)

from sympy.ntheory import count_digits
def ok(n): d = count_digits(n, 3); return 0 in d and 2 in d and d[0] == d[2]
print(list(filter(ok, range(285)))) # Michael S. Branicky, Jun 11 2021

A117564 Numbers with no 1's in base 3, 4 & 10 expansions.

Original entry on oeis.org

0, 2, 8, 56, 60, 62, 224, 234, 236, 240, 242, 546, 558, 560, 648, 650, 654, 672, 674, 702, 704, 40992, 40994, 43746, 43758, 43760, 43820, 43904, 43962, 43964, 43976, 43980, 43982, 44226, 44232, 44234, 44280, 44282, 44286, 45224, 45258, 45260, 45704

Offset: 1

Zak Seidov, Apr 29 2006

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

Cf. A005823, A023709, A117496.

a:=proc(n) if member(1,convert(convert(n,base,3),set) union convert(convert(n,base,4),set) union convert(convert(n,base,10),set))=false then n else fi end: seq(a(n),n=0..46000); # Emeric Deutsch, Oct 10 2006

n1Q[n_]:=DigitCount[n,3,1]==DigitCount[n,4,1]==DigitCount[n,10,1]==0; Select[Range[0,50000],n1Q] (* Harvey P. Dale, Sep 30 2012 *)

More terms from Emeric Deutsch, Oct 10 2006

A343754 a(n) = 0, and for any n > 0, a(n+1) = a(n) - A065363(n) + 1.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 3, 3, 4, 4, 3, 3, 2, 0, 3, 5, 6, 8, 9, 9, 10, 10, 9, 11, 12, 12, 13, 13, 12, 12, 11, 9, 10, 10, 9, 9, 8, 6, 5, 3, 0, 4, 7, 9, 12, 14, 15, 17, 18, 18, 21, 23, 24, 26, 27, 27, 28, 28, 27, 29, 30, 30, 31, 31, 30, 30, 29, 27, 30, 32, 33, 35, 36

Offset: 0

Rémy Sigrist, Apr 27 2021

This sequence has connections with A296062 and the Takagi (or blancmange) curve:
- for any real number x,
- let s(x) = min(frac(x), 1-frac(x)) (this is the building block of the Takagi curve),
- let t(x) = min(1/3, s(x)),
- let f(x) = Sum_{k >= 0} t(x * 3^k) / 3^k,
- the scatterplot of the sequence in the range A003462(k)..A003462(k+1)
  approaches the curve x -> f(x)*3^k for x in the range 0..1.

Rémy Sigrist, Table of n, a(n) for n = 0..9841
Rémy Sigrist, Scatterplot of the sequence for n = 0..A003462(11)
Rémy Sigrist, Representation of f in the range 0..1
Rémy Sigrist, Successive approximations of f
Wikipedia, Blancmange curve

Cf. A005823, A065363, A174574, A296062.

s=0; for (n=1, 73, print1 (s", "); m=n; while (m>1, s-=d=centerlift(Mod(m, 3)); m=(m-d)\3))

a(n) = n - A174574(n).
a(n) >= 0 with equality iff n belongs to A003462.
a(n) <= n/2 with equality iff n belongs to A005823.

A357616 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the ternary expansion of n equals the number of 2's in the ternary expansion of a(n) and vice versa.

Original entry on oeis.org

0, 2, 1, 6, 8, 5, 3, 7, 4, 18, 20, 11, 24, 26, 17, 15, 23, 14, 9, 19, 10, 21, 25, 16, 12, 22, 13, 54, 56, 29, 60, 62, 35, 33, 47, 32, 72, 74, 51, 78, 80, 53, 59, 71, 44, 45, 61, 34, 65, 77, 50, 38, 52, 41, 27, 55, 28, 57, 69, 42, 30, 46, 31, 63, 73, 48, 75, 79

Offset: 0

Rémy Sigrist, Oct 06 2022

This sequence is a self-inverse permutation of the nonnegative integers.

			The first terms, alongside their ternary expansions, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     2       1          2
   2     1       2          1
   3     6      10         20
   4     8      11         22
   5     5      12         12
   6     3      20         10
   7     7      21         21
   8     4      22         11
   9    18     100        200
  10    20     101        202
  11    11     102        102
  12    24     110        220

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A005823, A005836, A039001 (fixed points), A062756, A081603, A352760.

PARI
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See Links section.
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A081603(a(n)) = A062756(n).
A062756(a(n)) = A081603(n).
a(n) < 3^k iff n < 3^k.
a(n) = n iff n belongs to A039001.
Empirically:
- a(n) = n/2 iff n belongs to A005823,
- a(n) = 2*n iff n belongs to A005836.

A083904 G.f. 1/(1-x) * Sum_{k>=0} 3^k * x^2^(k+1)/(1+x^2^k).

Original entry on oeis.org

0, 1, 0, 4, 3, 1, 0, 13, 12, 10, 9, 4, 3, 1, 0, 40, 39, 37, 36, 31, 30, 28, 27, 13, 12, 10, 9, 4, 3, 1, 0, 121, 120, 118, 117, 112, 111, 109, 108, 94, 93, 91, 90, 85, 84, 82, 81, 40, 39, 37, 36, 31, 30, 28, 27, 13, 12, 10, 9, 4, 3, 1, 0, 364, 363, 361, 360

Offset: 1

Ralf Stephan, Jun 18 2003

Distance to next number of form 2^k-1, written down in binary, then interpreted as ternary. Thus the numbers have no 2 in ternary representation.

Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions.
Ralf Stephan, Table of generating functions

Cf. A005823, A005836.

for(n=1, 100, l=ceil(log(n)/log(2)); t=polcoeff(1/(1-x)*sum(k=0, l, 3^k*(x^2^(k+1))/(1+x^2^k)) + O(x^(n+1)), n); print1(t", "))

a(1)=0, a(2n) = 3a(n)+1, a(2n+1) = 3a(n).

a(n) = (1/2)*(3^(floor(log_2(n))+1)-1) - A005836(n).

A366344 Irregular triangle T(n, k), n >= 0, k = 1 or 2, read by rows; the n-th row contains two coprime positive integers whose prime factorizations are encoded in the ternary expansion of n (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 3, 2, 1, 3, 2, 3, 1, 4, 5, 1, 6, 1, 5, 2, 9, 1, 8, 1, 9, 2, 5, 3, 10, 3, 3, 4, 1, 5, 2, 5, 1, 6, 3, 5, 4, 3, 3, 10, 1, 9, 2, 9, 1, 8, 7, 1, 10, 1, 7, 2, 15, 1, 12, 1, 15, 2, 7, 3, 14, 3, 5, 4, 25, 1, 18, 1, 25, 2, 27, 1, 16, 1, 27, 2

Offset: 0

Rémy Sigrist, Oct 07 2023

The encoding used here is related to that used for the Doudna sequence (A005940):
- for any pair (u, v) of coprime positive integers, the ternary expansion of the unique n >= 0 such that T(n, 1) = u and T(n, 2) = v is built as follows (from right to left):
   - for m = 1, 2, ..., let p be the m-th prime number,
     - if p neither divides u nor v then we add a 0,
     - if p divides u with multiplicity e then we add a run of e 1's,
     - if p divides v with multiplicity e then we add a run of e 2's,
     - we also insert an extra 0 between pairs of runs of 1's not separated by 2's and between pairs of runs of 2's not separated by 1's.
This encoding can be applied to any fixed base b >= 2 and will yield a bijection from the nonnegative integers to the set of tuples of b-1 pairwise coprime positive integers.
The case b = 2 corresponds (up to the offset) to the Doudna sequence (A005940).
The sequence n -> T(n, 1) / T(n, 2) runs through all the reduced positive rationals exactly once.

			Triangle T(n, k) begins (alongside the ternary expansion of n):
  n   n-th row  ter(n)
  --  --------  ------
   0  [1, 1]         0
   1  [2, 1]         1
   2  [1, 2]         2
   3  [3, 1]        10
   4  [4, 1]        11
   5  [3, 2]        12
   6  [1, 3]        20
   7  [2, 3]        21
   8  [1, 4]        22
   9  [5, 1]       100
  10  [6, 1]       101
  11  [5, 2]       102
  12  [9, 1]       110
  13  [8, 1]       111
  14  [9, 2]       112
  15  [5, 3]       120
  16  [10, 3]      121
  17  [3, 4]       122

Cf. A000040, A001222, A003961, A004488, A005823, A005836, A005940, A062756, A081603, A160384.

row(n, b = 3) = { my (r = vector(b-1, d, 1), g = 0, t = 0); while (n, my (d = n % b); n \= b; g++; if (d, my (e = 1); while (n % b == d, e++; n \= b;); if (t==d, g--, t = d); r[d] *= prime(g)^e;);); return (r); }

T(n, 1) = 1 iff n belongs to A005823.
T(n, 2) = 1 iff n belongs to A005836.
T(A005836(n), 1) = A005940(n+1).
T(A005823(n), 2) = A005940(n+1).
A001222(T(n, 1)) = A062756(n).
A001222(T(n, 2)) = A081603(n).
A001222(T(n, 1) * T(n, 2)) = A160384(n).
T(A004488(n), 1) = T(n, 2).
T(A004488(n), 2) = T(n, 1).
T((3^e - 1)/2, 1) = 2^e for any e >= 0.
T(3^e - 1, 2) = 2^e for any e >= 0.
T(3^e, 1) = A000040(e + 1) for any e >= 0.
T(2 * 3^e, 2) = A000040(e + 1) for any e >= 0.
T(3*n, k) = A003961(T(n, k)).

cp's OEIS Frontend

A097260 Numbers whose set of base 14 digits is {0,D}, where D base 14 = 13 base 10.

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A190640 Numbers whose base-3 expansion ends in 2 and does not contain any 1's.

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A249102 Numbers with no 1's in base-7 expansion.

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A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

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A039276 Numbers whose base-3 representation has the same nonzero number of 0's and 2's.

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A117564 Numbers with no 1's in base 3, 4 & 10 expansions.

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A343754 a(n) = 0, and for any n > 0, a(n+1) = a(n) - A065363(n) + 1.

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A357616 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the ternary expansion of n equals the number of 2's in the ternary expansion of a(n) and vice versa.

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