A067576 -id:A067576 - cp's OEIS frontend

A356419 Inverse of A067576 considered as a permutation of the positive integers.

1, 2, 3, 4, 5, 8, 6, 7, 12, 17, 9, 23, 13, 18, 10, 11, 30, 38, 24, 47, 31, 39, 14, 57, 48, 58, 19, 69, 25, 32, 15, 16, 68, 80, 81, 93, 94, 108, 40, 107, 123, 139, 49, 156, 59, 70, 20, 122, 174, 193, 82, 213, 95, 109, 26, 234, 124, 140, 33, 157, 41, 50, 21, 22, 138, 155, 256

Offset: 1

Jianing Song, Aug 06 2022

			A067576(12) = 9, so a(9) = 12.

Cf. A067576, A000120, A068076, A263017, A067587.

a(n)=my(w=hammingweight(n), p=sum(i=1, n-1, hammingweight(i)==w)); binomial(w+p+1, 2) - p

from math import comb
def A356419(n):
    c, k = 0, 0
    for i,j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i,k)
    return comb(n.bit_count()+c+1,2)-c # Chai Wah Wu, Mar 02 2023

Let w(n) = A000120(n) be the Hamming weight of n, p(n) = A068076(n), then a(n) = binomial(w(n)+p(n)+1, 2) - p(n).

A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 21, 10, 13, 121, 77, 55, 27, 12, 17, 169, 143, 91, 65, 33, 14, 19, 289, 221, 187, 119, 85, 39, 16, 23, 361, 323, 247, 209, 133, 95, 45, 18, 29, 529, 437, 391, 299, 253, 161, 115, 51, 20, 31, 841, 667, 551, 493, 377, 319, 203, 125, 57, 22

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

A permutation of natural numbers >= 2.
The proportion of integers in the n-th row of the array is given by A005867(n-1)/A002110(n) = A038110(n)/A038111(n). - Peter Kagey, Jun 03 2019, based on comments by Jamie Morken and discussion with Tom Hanlon.
The proportion of the integers after the n-th row of the array is given by A005867(n)/A002110(n). - Tom Hanlon, Jun 08 2019

			Array begins:
   2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})
   3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)
   5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)
   7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)
  11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)
  13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)

Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.
Arrays of integers grouped into rows by various criteria:
by greatest prime factor: A125624,
by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),
by number of distinct prime factors: A125666,
by number of prime factors counted with multiplicity: A078840,
by prime signature: A095904,
by ordered prime signature: A096153,
by number of divisors: A119586,
by number of 1's in binary expansion: A066884 (upward), A067576 (downward),
by distance to next prime: A192179.
Cf. A002110, A005867, A038110, A038111.

a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]
(* second program: *)
rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* Jean-François Alcover, Mar 08 2016 *)

More terms from Hugo Pfoertner and Robert G. Wilson v, Jun 13 2003

A066884 Square array read by upward antidiagonals where the n-th row contains the positive integers with n binary 1's.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 15, 11, 6, 8, 31, 23, 13, 9, 16, 63, 47, 27, 14, 10, 32, 127, 95, 55, 29, 19, 12, 64, 255, 191, 111, 59, 30, 21, 17, 128, 511, 383, 223, 119, 61, 39, 22, 18, 256, 1023, 767, 447, 239, 123, 62, 43, 25, 20, 512, 2047, 1535, 895, 479, 247, 125, 79, 45, 26, 24, 1024

Offset: 1

Jared Benjamin Ricks (jaredricks(AT)yahoo.com), Jan 21 2002

This is a permutation of the positive integers; the inverse permutation is A067587.

			Column: 1   2   3   4   5   6
-----------------------------
Row 1:| 1   2   4   8  16  32
Row 2:| 3   5   6   9  10  12
Row 3:| 7  11  13  14  19  21
Row 4:|15  23  27  29  30  39
Row 5:|31  47  55  59  61  62
Row 6:|63  95 111 119 123 125

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)
Vladimir Dobric, M. Skyers, and L. J. Stanley, Polynomial Time Computable Triangular Arrays For Almost Sure Convergence, arXiv preprint arXiv:1603.04896 [math.PR], 2016. [Shows that this sequence is in P-TIME]
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A067587.
Selected rows: A000079 (1), A018900 (2), A014311 (3), A014312 (4), A014313 (5), A023688 (6), A023689 (7), A023690 (8), A023691 (9), A038461 (10), A038462 (11), A038463 (12). For decimal analogs, see A011557 and A038444-A038452.
Selected columns: A000225 (1), A055010 (2).
Selected diagonals: A036563 (main), A000918 (1st upper), A153894 (2nd upper). [Franklin T. Adams-Watters, Apr 22 2009]
Cf. A067576 (the same array read by downward antidiagonals).
Antidiagonal sums give A361074.

a = {}; Do[ a = Append[a, Last[ Take[ Take[ Select[ Range[2^12], Count[ IntegerDigits[ #, 2], 1] == j - i + 1 & ], j], i]]], {j, 1, 11}, {i, 1, j}]; a

Corrected and extended by Henry Bottomley, Jan 27 2002

A035073 a(n) is root of square starting with digit 6: first term of runs.

Original entry on oeis.org

8, 25, 78, 245, 775, 2450, 7746, 24495, 77460, 244949, 774597, 2449490, 7745967, 24494898, 77459667, 244948975, 774596670, 2449489743, 7745966693, 24494897428, 77459666925, 244948974279, 774596669242, 2449489742784, 7745966692415, 24494897427832, 77459666924149

Offset: 1

Patrick De Geest, Nov 15 1998

Subsequence of A045860.

Cf. A067576 (squares), A035076 (2..9).

from math import isqrt
def a(n): return isqrt(6*10**n) + 1
print([a(n) for n in range(1, 28)]) # Michael S. Branicky, Aug 25 2021

a(n) = ceiling(sqrt(6*10^n)), n > 0.

A207800 Permutation of positive numbers. See comments.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 5, 16, 15, 32, 6, 64, 11, 128, 9, 256, 31, 512, 10, 1024, 13, 2048, 12, 4096, 23, 8192, 17, 16384, 14, 32768, 18, 65536, 63, 131072, 20, 262144, 19, 524288, 24, 1048576, 27, 2097152, 33, 4194304, 21, 8388608, 34, 16777216, 47, 33554432, 36, 67108864, 22, 134217728, 40

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

nonn

a(1)=1; on places 2,4,6,8,... we put 2^m, m=1,2,3,..., i.e., numbers n with A000120(n)=1; on places 3,7,11,15,... we put numbers n with A000120(n)=2; on places 5,13,21,29,... we put numbers n with A000120(n)=3; etc.

For general description of the order, see comment in A207790.

Cf. A207790.

a(n) = A066884(A209268(A065190(n))). Equivalently, a(n) = A067576(A249725(A065190(n))). - Ivan Neretin, Apr 30 2016

a(30) corrected by Ivan Neretin, Apr 30 2016

A361074 Sum of the j-th number with binary weight n-j+1 over all j in [n].

Original entry on oeis.org

0, 1, 5, 16, 40, 92, 193, 401, 812, 1632, 3261, 6526, 13030, 26049, 52013, 103974, 207797, 415496, 830636, 1661086, 3321498, 6642591, 13283920, 26567121, 53131653, 106261922, 212518857, 425034976, 850060303, 1700115399, 3400211408, 6800412866, 13600787296

Offset: 0

Alois P. Heinz, Mar 01 2023

			a(0) = 0 (empty sum).
a(1) = 1 = 1_2.
a(2) = 5 = 2 + 3 = 10_2 + 11_2.
a(3) = 16 = 4 + 5 + 7 = 100_2 + 101_2 + 111_2.
a(4) = 40 = 8 + 6 + 11 + 15 = 1000_2 + 110_2 + 1011_2 + 1111_2.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Antidiagonal sums of A066884 or of A067576.

Cf. A000120, A057168.

b:= proc(i, j) option remember; uses Bits: local c, l, k;
      if j=1 then 2^i-1
    else c, l:= 0, [Split(b(i, j-1))[], 0];
         for k while l[k]<>1 or l[k+1]<>0 do c:=c+l[k] od;
         Join([1$c, 0$k-c, 1, l[k+2..-1][]])
      fi
    end:
a:= n-> add(b(j, n-j+1), j=1..n):
seq(a(n), n=0..32);

a(n) = Sum_{j=1..n} A066884(j,n-j+1) = Sum_{j=1..n} A067576(j,n-j+1).

Conjecture: a(n) ~ 19 * 2^n / 6. - Vaclav Kotesovec, Mar 04 2023

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.

Original entry on oeis.org

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

A134562 Array T by antidiagonals: T(n,k) = k-th number whose formal base-3 representation has exactly n terms. ("Formal" means that all the nonzero coefficients are 1's.).

Original entry on oeis.org

1, 3, 2, 9, 4, 5, 27, 6, 7, 8, 81, 10, 11, 14, 17, 243, 12, 13, 16, 23, 26, 729, 18, 15, 20, 25, 44, 53, 2187, 28, 19, 22, 35, 50, 71

Offset: 1

Clark Kimberling, Nov 01 2007

A permutation of the natural numbers. Except for initial terms in some cases, (Row 1) = A000244, (Row 2) = A055235, (Col 1) = A062318. For the analogous base-2 array, see A067576.

			11 = 9 + 1 + 1 is the 3rd largest number (after 5 and 7) that has
a 3-term formal base-3 representation.
Northwest corner:
1 3 9 27 81
2 4 6 10 12
5 7 11 13 15
8 14 16 20 22

Cf. A067576, A000244, A055235, A062318.

cp's OEIS Frontend

A356419 Inverse of A067576 considered as a permutation of the positive integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A066884 Square array read by upward antidiagonals where the n-th row contains the positive integers with n binary 1's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A035073 a(n) is root of square starting with digit 6: first term of runs.

Views

Author

Keywords

Crossrefs

Programs

Python

Formula

A207800 Permutation of positive numbers. See comments.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A361074 Sum of the j-th number with binary weight n-j+1 over all j in [n].

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A134562 Array T by antidiagonals: T(n,k) = k-th number whose formal base-3 representation has exactly n terms. ("Formal" means that all the nonzero coefficients are 1's.).

Views

Author

Keywords

Comments

Examples

Crossrefs