A018900 -id:A018900 - cp's OEIS frontend

A086772 Store the natural numbers in a triangular array such that values on each row have the same number of bits. Start a new row with the smallest number not yet recorded. a(n) represents the initial terms in the resulting array.

0, 1, 3, 4, 7, 9, 15, 21, 24, 31, 41, 45, 63, 64, 72, 74, 83, 94, 127, 139, 140, 173, 197, 207, 234, 255, 268, 284, 288, 339, 349, 390, 426, 445, 467, 511, 522, 553, 569, 634, 689, 706, 734, 797, 838, 934, 950, 951, 1023, 1036, 1052, 1078, 1179, 1236

Offset: 0

Alford Arnold, Aug 03 2003

A067576 describes the sequences with a fixed number of binary bits using antidiagonals.

			The array begins:
   0
   1  2
   3  5  6
   4  8 16 32
   7 11 13 14 19
   9 10 12 17 18 20
  15 23 27 29 30 39 43
  ...
so the initial terms are 0 1 3 4 7 9 15 ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000079, A001477, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691.

A086772aux := proc(n,k)
    option remember;
    local a,npr,kpr,fnd ;
    if n = 0 then
        return 0;
    end if;
    if k = 0 then
        for a from 1 do
            fnd := false;
            for npr from 1 to n-1 do
                for kpr from 0 to npr do
                    if procname(npr,kpr) = a then
                        fnd := true;
                        break;
                    end if;
                end do:
            end do:
            if not fnd then
                return a;
            end if;
        end do:
    else
        for a from 1 do
            if wt(a) = wt(procname(n,0)) then
                fnd := false;
                for npr from 1 to n-1 do
                    for kpr from 0 to npr do
                        if procname(npr,kpr) = a then
                            fnd := true;
                            break;
                        end if;
                    end do:
                end do:
                for kpr from 0 to k-1 do
                    if procname(n,kpr) = a then
                        fnd := true;
                        break;
                    end if;
                end do:
                if not fnd then
                    return a;
                end if;
            end if;
        end do:
    end if;
end proc:
A086772 := proc(n)
    A086772aux(n,0) ;
end proc: # R. J. Mathar, Sep 15 2012

A340161 a(n) is the smallest number k for which the set {k + 1, k + 2, ..., k + k} contains exactly n elements with exactly three 1-bits (A014311).

Original entry on oeis.org

1, 4, 6, 7, 10, 11, 13, 18, 19, 21, 25, 34, 35, 37, 41, 49, 66, 67, 69, 73, 81, 97, 130, 131, 133, 137, 145, 161, 193, 258, 259, 261, 265, 273, 289, 321, 385, 514, 515, 517, 521, 529, 545, 577, 641, 769, 1026, 1027, 1029, 1033, 1041, 1057, 1089, 1153, 1281, 1537

Offset: 0

Marius A. Burtea, Dec 30 2020

When n = m*(m-1)/2 + 1, m >= 2 (A000124 \ {1}), then a(n) = k = 2^m+2, m >= 2 (A052548 \ {3, 4}), and only for these values of k, there exists only one set, {k+1, k+2, ..., 2k}, that contains exactly n elements whose binary representation has exactly three 1's (see A340068). - Bernard Schott, Jan 03 2021
From David A. Corneth, Jan 03 2021: (Start)
a(n) = A018900(n) + 1 for n >= 1.
Proof: Let T(k) be the number of values in {k+1, k+2, ..., k+k} that have exactly 3 ones in their binary expansion. Let h(k) be 1 if k has exactly 3 ones in its binary expansion and 0 otherwise and let w(k) be the binary weight of k (cf. A000120). Then T(k + 1) = T(k) + h(2*k + 1) + h(2*k + 2) - h(k + 1) = T(k) + h(2*k + 1) + h(2*(k + 1)) - h(k + 1) but as h(2^m * k) = h(k) two terms cancel and we have T(k + 1) = T(k) + h(2*k + 1). If w(2*k + 1) = w(k) + 1 = 3 then w(k) = 2 which holds for k in A018900. (End)

			For k in {1, 2, 3}, the sets are {1, 2}, {3, 4} and {4, 5, 6}, which do not contain numbers in A014311, so a(0) = 1.
For k = 4, the set is {5, 6, 7, 8} with 7 = A014311(1), so a(1) = 4.
For k = 6, the set {7, 8, 9, 10, 11, 12} contains the elements 7 = A014311(1) and 11 = A014311(2), so a(2) = 6.

Essentially a duplicate of A018900 - N. J. A. Sloane, Jan 23 2021.
Cf. A014311, A340068.
Cf. also A000124, A052548 \ {3} (is a subsequence).

fb:=func; a:=[]; for n in [0..64] do k:=1; while #[s:s in [k+1..2*k]|fb(s)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

first(n) = {my(res = vector(n), t = 1); res[1] = 1; for(i = 2, oo, if(hammingweight(2*i-1) == 3, t++; if(t > n, return(res)); res[t] = i))} \\ David A. Corneth, Jan 03 2021

from math import isqrt, comb
def A340161(n): return 1+(1<<(m:=isqrt(n<<3)+1>>1))+(1<<(n-1-comb(m,2))) if n else 1 # Chai Wah Wu, Mar 10 2025

From Bernard Schott, Jan 03 2021: (Start)
a(m*(m-1)/2 + 1) = 2^m + 2 for m >= 2.
a(m*(m-1)/2 + 2) = 2^m + 3 for m >= 2.
a(n) = A018900(n) + 1 for n >= 1 (see A340068). (End)

A356354 a(n) is the least k such that the sets of positions of 1's in the binary expansions of n and k are similar.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 11, 3, 11, 7, 15, 1, 3, 3, 19, 3, 7, 11, 23, 3, 19, 11, 27, 7, 23, 15, 31, 1, 3, 3, 35, 3, 37, 19, 39, 3, 37, 7, 43, 11, 45, 23, 47, 3, 35, 19, 51, 11, 43, 27, 55, 7, 39, 23, 55, 15, 47, 31, 63, 1, 3, 3, 67, 3, 11, 35, 71, 3, 7

Offset: 0

Rémy Sigrist, Oct 15 2022

Let s(n) be the set of terms in the n-th row of A133457 (with s(0) = {}).

a(n) is the least k such that s(n) is the image of s(k) under some nonconstant linear function.

			The first terms, alongside their binary expansions, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1      10          1
   3     3      11         11
   4     1     100          1
   5     3     101         11
   6     3     110         11
   7     7     111        111
   8     1    1000          1
   9     3    1001         11
  10     3    1010         11
  11    11    1011       1011
  12     3    1100         11
  13    11    1101       1011
  14     7    1110        111
  15    15    1111       1111
  16     1   10000          1

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program
Wikipedia, Similarity (geometry)
Index entries for sequences related to binary expansion of n

Cf. A000120, A000265, A018900, A030101, A064895, A133457.

PARI
```
See Links section.
```

A000120(a(n)) = A000120(n).
a(a(n)) = a(n).
a(2*n) = a(n).
a(A030101(n)) = a(n).
a(n) = 1 iff n is a power of 2.
a(n) = 3 iff n belongs to A018900.
a(2^k - 1) = 2^k - 1 for any k >= 0.

A078459 Numbers n such that in some base b <= log n the sum of the digits of n in base b equals b.

Original entry on oeis.org

9, 10, 12, 17, 18, 20, 21, 24, 29, 31, 33, 34, 36, 37, 39, 40, 45, 48, 55, 57, 63, 65, 66, 67, 68, 70, 72, 73, 76, 80, 82, 83, 85, 87, 88, 91, 93, 96, 97, 99, 100, 109, 111, 112, 117, 129, 130, 132, 133, 135, 136, 144, 145, 148, 153, 157, 160, 161, 163, 165, 171, 177

Offset: 1

Carlos Alves, Dec 31 2002

The sequence A018900 refers to b=2. Other sequences with b=3,4,5,... could be considered separately. We present the case that includes the several b, provided that b<=Log(n) (this condition avoids trivial situations). - It is not completely clear how the sequence grows.

			In base 4, 67=(1003)_4 and the sum 1+0+0+3 equals 4.

In a sense this generalizes the sequence A018900 (if b=2 is considered).

Seq = Function[{m}, Module[{ls, sk}, ls = {}; Do[Do[ sk = Apply[Plus, IntegerDigits[k, b]]; If[sk == b && E^b <= k, AppendTo[ls, k]], {b, 2, Log[k]}], {k, 1, m}]; Union[Sort[ls]] ]]; Seq[1000]

A091860 a(1)=1, a(n)=sum(i=1,n-1,b(i)) where b(i)=0 if a(i) and a(n-i) are both even, b(i)=1 otherwise.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 5, 6, 6, 8, 6, 8, 8, 8, 7, 8, 8, 10, 8, 10, 10, 10, 8, 10, 10, 10, 10, 10, 10, 10, 9, 10, 10, 12, 10, 12, 12, 12, 10, 12, 12, 12, 12, 12, 12, 12, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 12, 12, 14, 12, 14, 14, 14, 12, 14, 14, 14, 14, 14

Offset: 1

Benoit Cloitre, Mar 16 2004

nonn

Cf. A018900, A072823.

an[1]=1;for(n=2,100,an[n]=sum(i=1,n-1,max(a(i)%2,a(n-i)%2)))

n>1 a(2n)=a(n)+2; if n is a power of 2, a(2n+1)=1+a(2n); if n is the sum of 2 distinct power of 2, a(2n+1)=2+a(2n); a(2n+1)=a(2n) otherwise

A139369 Array read by antidiagonals, n-th sum of 2 distinct powers of k.

Original entry on oeis.org

3, 4, 5, 5, 10, 6, 6, 17, 12, 9, 7, 26, 20, 28, 10, 8, 37, 30, 65, 30, 12, 9, 50, 42, 126, 68, 36, 17, 10, 65, 56, 217, 130, 80, 82, 18, 11, 82, 72, 344, 222, 150, 257, 84, 20, 12, 101, 90, 513, 350, 252, 626, 260, 90, 24, 13, 122, 110, 730, 520, 392, 1297, 630, 272, 108

Offset: 1

Jonathan Vos Post, Jun 07 2008

n=2 column is A002522 n^2 + 1.

n=3 column is A002378 n*(n+1) Oblong (or pronic, promic, or heteromecic numbers).

			Array begins:
================================================================================
k....|.n=1.|.n=2.|.n=3.|..n=4.|..n=5.|..n=6.|...n=7.|...n=8.|..n=9.|.n=10|.OEIS.
================================================================================
k=2..|..3..|...5.|..6..|....9.|...10.|...12.|....17.|...18..|...20.|..24.|A018900
k=3..|..4..|..10.|.12..|...28.|...30.|...36.|....82.|...84..|...90.|..108|A038464
k=4..|..5..|..17.|.20..|...65.|...68.|...80.|...257.|..260..|..272.|..320|A038470
k=5..|..6..|..26.|.30..|..126.|..130.|..150.|...626.|..630..|..650.|..750|A038474
k=6..|..7..|..37.|.42..|..217.|..222.|..252.|..1297.|..1302.|.1332.|.1512|A038478
k=7..|..8..|..50.|.56..|..344.|..350.|..392.|..2402.|..2408.|.2450.|.2744|A038481
k=8..|..9..|..65.|.72..|..513.|..520.|..576.|..4097.|..4104.|.4160.|.4608|A038484
k=9..|.10..|..82.|.90..|..730.|..738.|..810.|..6562.|..6570.|.6642.|.7290|A038487
k=10.|.11..|.101.|.110.|.1001.|.1010.|.1100.|.10001.|.10010.|10100.|11000|A038444
k=11.|.12..|.122.|.132.|.1332.|.1342.|.1452.|.14642.|.14652.|14762.|15972|A038490
k=12.|.13..|.145.|.156.|.1729.|.1740.|.1872.|.20737.|.20748.|20880.|22464|A038492
================================================================================

Cf. A002378, A002522, A018900, A038464, A038470, A038474, A038478, A038481, A038484, A038487, A038444, A038490, A038492.

A367680 Number of integer compositions x1+x2+...+xk of n such that each xj has exactly j bits set.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 1, 3, 2, 1, 2, 4, 2, 4, 6, 2, 4, 5, 10, 7, 10, 12, 8, 6, 11, 14, 16, 13, 16, 16, 14, 14, 30, 32, 19, 35, 28, 23, 27, 38, 36, 47, 44, 42, 55, 52, 51, 85, 88, 74, 84, 84, 72, 81, 102, 110, 122, 115, 108, 132, 137, 136, 179, 195, 164, 160, 181

Offset: 0

Arnauld Chevallier, Nov 26 2023

			There are 6 such compositions for n = 14:
  14 = 1 + 6 + 7 (1 + 110 + 111)
  14 = 2 + 5 + 7 (10 + 101 + 111)
  14 = 2 + 12 (10 + 1100)
  14 = 4 + 3 + 7 (100 + 11 + 111)
  14 = 4 + 10 (100 + 1010)
  14 = 8 + 6 (1000 + 110)
Therefore a(14) = 6.

Cf. A000079, A000120, A018900, A014311, A014313, A023688, A023689, A023690, A023691.

a(n) = my(nb=0); forpart(v=n, if (vecsort(apply(hammingweight, Vec(v))) == [1..#v], nb++)); nb; \\ Michel Marcus, Nov 28 2023

cp's OEIS Frontend

A086772 Store the natural numbers in a triangular array such that values on each row have the same number of bits. Start a new row with the smallest number not yet recorded. a(n) represents the initial terms in the resulting array.

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Maple

A340161 a(n) is the smallest number k for which the set {k + 1, k + 2, ..., k + k} contains exactly n elements with exactly three 1-bits (A014311).

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A356354 a(n) is the least k such that the sets of positions of 1's in the binary expansions of n and k are similar.

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PARI

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A078459 Numbers n such that in some base b <= log n the sum of the digits of n in base b equals b.

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Mathematica

A091860 a(1)=1, a(n)=sum(i=1,n-1,b(i)) where b(i)=0 if a(i) and a(n-i) are both even, b(i)=1 otherwise.

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PARI

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A139369 Array read by antidiagonals, n-th sum of 2 distinct powers of k.

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A367680 Number of integer compositions x1+x2+...+xk of n such that each xj has exactly j bits set.

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PARI