A033042 -id:A033042 - cp's OEIS frontend

A215088 a(n)=Sum{d(i)2^i: i=0,1,...,m}, where Sum{d(i)5^i: i=0,1,...,m} is the base 5 representation of n.

0, 1, 2, 3, 4, 2, 3, 4, 5, 6, 4, 5, 6, 7, 8, 6, 7, 8, 9, 10, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 6, 7, 8, 9, 10, 8, 9, 10, 11, 12, 10, 11, 12, 13, 14, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 10, 11, 12, 13, 14, 12, 13, 14, 15, 16, 14, 15, 16, 17, 18, 16, 17, 18, 19, 20, 12, 13, 14, 15, 16, 14, 15, 16, 17, 18, 16, 17, 18, 19, 20, 18, 19, 20, 21, 22, 20, 21, 22, 23

Offset: 0

Clark Kimberling, Aug 02 2012

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A033042.

t = Table[FromDigits[RealDigits[n, 5], 2], {n, 0, 100}]

a(n) = 2*a(n/5) if n = 0 mod 5; otherwise, a(n) = a(n-1) + 1. - Clark Kimberling, Aug 03 2012

A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 1, 3, 3, 1, 0, 0, 2, 4, 4, 4, 1, 0, 1, 2, 5, 9, 5, 5, 1, 0, 0, 3, 6, 10, 16, 6, 6, 1, 0, 1, 1, 7, 12, 17, 25, 7, 7, 1, 0, 0, 2, 8, 13, 20, 26, 36, 8, 8, 1, 0, 1, 2, 9, 27, 21, 30, 37, 49, 9, 9, 1, 0, 0, 3, 10, 28, 64, 31, 42, 50, 64, 10, 10, 1, 0

Offset: 0

Rémy Sigrist, Jun 04 2021

For any n >= 0, the n-th row, k -> T(n, k), corresponds to a polynomial in k with coefficients in {0, 1}.

For any k > 1, the k-th column, n -> T(n, k), corresponds to sums of distinct powers of k.

			Array T(n, k) begins:
  n\k|  0  1   2   3   4    5    6    7    8    9    10    11    12
  ---+-------------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0     0     0     0
    1|  1  1   1   1   1    1    1    1    1    1     1     1     1
    2|  0  1   2   3   4    5    6    7    8    9    10    11    12
    3|  1  2   3   4   5    6    7    8    9   10    11    12    13
    4|  0  1   4   9  16   25   36   49   64   81   100   121   144
    5|  1  2   5  10  17   26   37   50   65   82   101   122   145
    6|  0  2   6  12  20   30   42   56   72   90   110   132   156
    7|  1  3   7  13  21   31   43   57   73   91   111   133   157
    8|  0  1   8  27  64  125  216  343  512  729  1000  1331  1728
    9|  1  2   9  28  65  126  217  344  513  730  1001  1332  1729
   10|  0  2  10  30  68  130  222  350  520  738  1010  1342  1740
   11|  1  3  11  31  69  131  223  351  521  739  1011  1343  1741
   12|  0  2  12  36  80  150  252  392  576  810  1100  1452  1872

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Index entries for sequences related to binary expansion of n

See A342707 for a similar sequence.

Cf. A000035, A000120, A000583, A000695, A001093, A002061, A002378, A002522, A002523, A005836, A007088, A011379, A027444, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A034262, A053698, A071568, A098547, A104258.

T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^e); v }

T(n, n) = A104258(n).
T(n, 0) = A000035(n).
T(n, 1) = A000120(n).
T(n, 2) = n.
T(n, 3) = A005836(n).
T(n, 4) = A000695(n).
T(n, 5) = A033042(n).
T(n, 6) = A033043(n).
T(n, 7) = A033044(n).
T(n, 8) = A033045(n).
T(n, 9) = A033046(n).
T(n, 10) = A007088(n).
T(n, 11) = A033047(n).
T(n, 12) = A033048(n).
T(n, 13) = A033049(n).
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(4, k) = k^2.
T(5, k) = k^2 + 1 = A002522(k).
T(6, k) = k^2 + k = A002378(k).
T(7, k) = k^2 + k + 1 = A002061(k).
T(8, k) = k^3.
T(9, k) = k^3 + 1 = A001093(k).
T(10, k) = k^3 + k = A034262(k).
T(11, k) = k^3 + k + 1 = A071568(k).
T(12, k) = k^3 + k^2 = A011379(k).
T(13, k) = k^3 + k^2 + 1 = A098547(k).
T(14, k) = k^3 + k^2 + k = A027444(k).
T(15, k) = k^3 + k^2 + k + 1 = A053698(k).
T(16, k) = k^4 = A000583(k).
T(17, k) = k^4 + 1 = A002523(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

Original entry on oeis.org

1, 2, 20, 280, 4620, 84084, 1633632, 33256080, 701149020, 15191562100, 336424047960, 7584833081280, 173575987821600, 4022766574898400, 94247674040476800, 2228957491057276320, 53150802525726081660, 1276661433215969608500, 30863850087221160009000

Offset: 0

Paul D. Hanna, Oct 10 2023

Equals the central terms of triangle A365770.
Conjectures: given A033042 is the sums of distinct powers of 5, then
(1) a(5*A033042(n)) == 4 (mod 5) for n > 0,
(2) a(5*A033042(n) + 1) == 2 (mod 5) for n > 0,
(3) a(n) == 0 (mod 5) for n > 0 except when n or n-1 equals 5*A033042(k) for some k >= 0.

Paolo Xausa, Table of n, a(n) for n = 0..500

Cf. A007004, A000108, A165817, A033042, A319578, A356770.

A365771[n_] := Binomial[2*n + 1, n]/(2*n + 1)*Binomial[3*n - 1, n];
Array[A365771, 20, 0] (* Paolo Xausa, Oct 12 2024 *)

{a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n)}
for(n=0,30,print1(a(n),", "))

from math import comb
def A365771(n): return comb(m:=(n<<1)+1,n)*comb(m+n-2,n)//m if n else 1 # Chai Wah Wu, Oct 11 2023

a(n) = A365770(2*n,n) for n >= 0.
a(n) = A000108(n) * A165817(n) for n >= 0.
a(n) = 2*A319578(n) = (2/3) * A007004(n) for n >= 1. - Peter Bala, Aug 25 2025

A037410 Positive numbers having the same set of digits in base 2 and base 5.

Original entry on oeis.org

1, 5, 25, 26, 30, 31, 125, 126, 130, 131, 150, 151, 155, 625, 626, 630, 631, 650, 651, 655, 656, 750, 751, 755, 756, 775, 776, 780, 3125, 3126, 3130, 3131, 3150, 3151, 3155, 3156, 3250, 3251, 3255, 3256, 3275, 3276, 3280, 3281

Offset: 1

Clark Kimberling

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A033042.

Select[Range[3500],Union[IntegerDigits[#,2]]==Union[IntegerDigits[#,5]]&] (* Harvey P. Dale, May 08 2025 *)

isok(n) = Set(digits(n, 2)) == Set(digits(n, 5)); \\ Michel Marcus, Jan 11 2017

A258946 Numbers that can be expressed using only the digits 0 and 1 in no more than three different bases.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 11, 14, 15, 18, 19, 22, 23, 24, 29, 32, 33, 34, 35, 38, 41, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 66, 67, 70, 71, 74, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 92, 95, 96, 97, 98, 99, 102, 103, 104, 105, 106, 107

Offset: 1

Thomas Oléron Evans, Jun 15 2015

All integers n >= 4 may trivially be expressed using only the digits 0 and 1 in three different bases: 2, n-1 (as '11') and n (as '10'). The numbers in this sequence cannot be expressed using only 0 and 1 in any other base.
The only positive integers that may be expressed using only the digits 0 and 1 in fewer than three different bases are 2 and 3, for which the values {2, n-1, n} are not all distinct or are not all valid bases.
An equivalent definition: For each term a(n) of this sequence, there are at most three integers k >= 2 for which a(n) is a sum of distinct nonnegative integer powers of k.

			5 is a term of the sequence, because 5 may be expressed using only the digits 0 and 1 in precisely three different bases: 2, 4 and 5 (5 is '12' in base 3).
9 is not a term of the sequence, because 9 can be expressed using only the digits 0 and 1 in four different bases: 2, 3, 8, 9 (9 is '100' in base 3).

Thomas Oléron Evans, Table of n, a(n) for n = 1..10000
Thomas Oléron Evans, Python program

Subsequence of A074940.

Cf. A146025, A258107, A005836, A000695, A033042.

filter:= proc(n)
  local b;
  for b from 3 to n-2 do
    if max(convert(n,base,b)) <= 1 then return false
    fi
  od:
true
end proc:
select(filter, [$2..1000]); # Robert Israel, Jun 19 2015

is(n)=if(n<2, return(0)); for(b=3,sqrtint(n),if(vecmax(digits(n,b))<2, return(0))); 1 \\ Charles R Greathouse IV, Jun 15 2015

A268337 Numbers which have only digits 0 and 1 in bases 3 and 5.

Original entry on oeis.org

0, 1, 30, 31, 756, 3250, 3276, 3280, 81255, 81256, 81280, 81900, 81901, 82000, 59078250, 59078251, 59078280, 59078281, 31789468750, 31789468776, 31789469505, 31789469506, 31789471900, 31789471905, 31789471906, 31789472005, 946095722031, 946095800025, 946095800026, 946095800031, 946095800130

Offset: 1

M. F. Hasler, Feb 01 2016

The number 82000 is famous for having only digits 0 and 1 in all bases <= 5, no other such number > 1 is known. See also A146025 and A258981.
If explicit formulas for (convenient) infinite subsequences of this one can be found, this could open new ways to progress on this problem.
The terms come in groups having roughly the first half (or at least third) of digits in common, see the link "Terms in base 10, 5 and 3".

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 84 terms from M. F. Hasler, next 31 terms from Charles R Greathouse IV)
M. F. Hasler, Terms in base 10, 5 and 3.

Cf. A005836, A033042.

d:= 20: # to get all terms < 5^d
res:= NULL:
T:= combinat:-cartprod([[$0..1]$d]):
while not T[finished] do
  r:= T[nextvalue]();
  v:= add(r[i]*5^(d-i),i=1..d);
  if max(convert(v,base,3)) <= 1 then
    res:= res,v
  fi
od:
res; # Robert Israel, Feb 01 2016

Module[{t=Tuples[{0,1},25],b3,b5},b3=FromDigits[#,3]&/@t;b5=FromDigits[ #,5]&/@t;Intersection[b3,b5]] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Dec 13 2021 *)

print1(0);for(n=1,1e10,vecmax(digits(t=subst(Pol(binary(n)),'x,5),3))<2&&print1(","t))

list(lim)=my(v=List([0]),d=digits(lim\1,5),t); for(i=1,#d, if(d[i]>1, for(j=i,#d, d[j]=1); break)); for(n=1,fromdigits(d,5), t=fromdigits(binary(n),5); if(vecmax(digits(t,3))<2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 02 2016

a(n) >> n^k with k = log 5/log 2 = 2.321928.... - Charles R Greathouse IV, Feb 02 2016

A263684 Numbers whose base-4 and base-5 representations have only 0's and 1's.

Original entry on oeis.org

0, 1, 5, 16400, 16401, 16405, 82000, 82001, 82005

Offset: 1

Robert Israel, Oct 23 2015

Intersection of A000695 and A033042.

These appear to be all the terms. There are no more below 10^500.

			16400 is 10000100 in base 4 and 1011100 in base 5.

Cf. A000695, A033042, A146025, A258981.

split:= proc(ab, B)
   local a,b,La, Lb, k, j, a1,  a2, b1, b2, x;
   global Res, count;
   a:= ab[1]; b:= ab[2];
   if b-a <= 1000 then
      for x from a to b-1 do
        if max(convert(x,base,4)) <= 1 and max(convert(x,base,5)) <= 1 then
           count:= count+1; Res[count]:= x
        fi
      od;
      return ({});
   fi;
   La:= convert(a,base,B);
   Lb:= convert(b,base,B);
   if nops(Lb) > nops(La) then La:= [op(La),0$(nops(Lb)-nops(La))] fi;
   k:= ListTools:-SelectLast(`>`,Lb-La,0,output=indices);
   if La[k] = 0 then
     a1:= a;
     b1:= 2 + add(B^i,i=1..k-2) + add(La[i]*B^(i-1),i=k+1..nops(La));
     a2:= B^(k-1) + add(La[i]*B^(i-1),i=k+1..nops(La));
     b2:= min(b, b1 + B^(k-1));
     return(select(t -> (t[1] t[1] 0 do
   Cands:= map(op@split, Cands, 5);
   Cands:= map(op@split, Cands, 4);
od:
sort(convert(Res,list));

Select[Range[0,83000],Max[Join[IntegerDigits[#,4],IntegerDigits[#,5]]]<2&] (* Harvey P. Dale, Sep 04 2018 *)

isok(n) = (n==0) || ((vecmax(digits(n,4))<=1) && (vecmax(digits(n,5))<=1)); \\ Michel Marcus, Oct 24 2015

cp's OEIS Frontend

A215088 a(n)=Sum{d(i)*2^i: i=0,1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.

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A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

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A365771 a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n) for n >= 0.

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A037410 Positive numbers having the same set of digits in base 2 and base 5.

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Mathematica

PARI

A258946 Numbers that can be expressed using only the digits 0 and 1 in no more than three different bases.

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Maple

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A268337 Numbers which have only digits 0 and 1 in bases 3 and 5.

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Maple

Mathematica

PARI

PARI

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A263684 Numbers whose base-4 and base-5 representations have only 0's and 1's.

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Maple

Mathematica

PARI

A215088 a(n)=Sum{d(i)2^i: i=0,1,...,m}, where Sum{d(i)5^i: i=0,1,...,m} is the base 5 representation of n.

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.