A340069 -id:A340069 - cp's OEIS frontend

A340100 Fixed points of A340069.

0, 1, 2, 3, 4, 6, 730, 1449, 4803, 8506, 10837, 17012, 18321, 21674, 43348, 86696, 103622, 103628, 153696, 173392, 185395, 207244, 227393, 370780, 454786, 693568, 936337, 989572, 1172353, 1284865, 1387136, 1413399, 1820929, 1872674, 2344706, 2569730, 2774272, 2826798

Offset: 1

Thomas Scheuerle, Dec 28 2020

A subset of A077436 because A000120(a(n)) = A000120(a(n)^2).

Cf. A340069, A077436, A000120.

a(n) = A340069(a(n)).

Incorrect terms removed and a(16)-a(33) added by Chai Wah Wu, Jan 08 2021

a(34)-a(38) from Chai Wah Wu, Jan 14 2021

A340351 Square array, read by descending antidiagonals, where row n gives all solutions k > 0 to A000120(k)=A000120(k*n), A000120 is the Hamming weight.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 4, 3, 6, 1, 5, 4, 7, 2, 7, 6, 5, 12, 3, 14, 3, 7, 6, 14, 4, 15, 6, 7, 8, 7, 15, 5, 27, 7, 14, 1, 9, 8, 24, 6, 28, 12, 15, 2, 15, 10, 9, 28, 7, 30, 14, 19, 3, 30, 7, 11, 10, 30, 8, 31, 15, 28, 4, 31, 14, 3, 12, 11, 31, 9, 39, 24, 30, 5, 43, 15, 6, 3, 13, 12

Offset: 1

Thomas Scheuerle, Jan 05 2021

Square array is read by descending antidiagonals, as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Rows at positions 2^k are 1, 2, 3, ..., (A000027). Row 2n is equal to row n.
Values are different from those in A115872, because we use multiplication with carry here.

			Eight initial terms of rows 1 - 8 are listed below:
   1:  1,  2,  3,   4,   5,   6,   7,   8, ...
   2:  1,  2,  3,   4,   5,   6,   7,   8, ...
   3:  3,  6,  7,  12,  14,  15,  24,  28, ...
   4:  1,  2,  3,   4,   5,   6,   7,   8, ...
   5:  7, 14, 15,  27,  28,  30,  31,  39, ...
   6:  3,  6,  7,  12,  14,  15,  24,  28, ...
   7:  7, 14, 15,  19,  28,  30,  31,  37, ...
   8:  1,  2,  3,   4,   5,   6,   7,   8, ...
a(6,3) = 7 because: 7 in binary is 111 and 6*7 = 42 in binary is 101001, both have 3 bits set to 1.

Cf. A000120, A292849 (1st column), A340069, A077459 (3rd row).

function [a] = A340351(max_n)
    for n = 1:max_n
        m = 1;
        k = 1;
        while m < max_n
            c = length(find(bitget(k,1:32)== 1));
            if c == length(find(bitget(n*k,1:32)== 1))
                a(n,m) = k;
                m = m+1;
            end
            k = k +1;
        end
    end
end

A340441 Square array, read by ascending antidiagonals, where row n gives all odd solutions k > 1 and n > 0 to A000120(2n+1) = A000120((2n+1)*k), A000120 is the Hamming weight.

Original entry on oeis.org

3, 13, 11, 3, 205, 43, 57, 5, 3277, 171, 35, 3641, 7, 52429, 683, 21, 47, 233017, 19, 838861, 2731, 3, 79, 99, 14913081, 23, 13421773, 10923, 241, 5, 197, 187, 954437177, 37, 214748365, 43691, 7, 61681, 7, 325, 419, 61083979321, 39, 3435973837, 174763

Offset: 1

Thomas Scheuerle, Jan 07 2021

Solutions to related equation A000120(k) = A000120(k*n) are A340351.

			Five initial terms of rows 1-5 are listed below:
   1:  3,   11,     43,       171,       683, ...
   2: 13,  205,   3277,     52429,    838861, ...
   3:  3,    5,      7,        19,        23, ...
   4: 57, 3641, 233017,  14913081, 954437177, ...
   5: 35,   47,      99,      187,       419, ...
T(3,4) = 19 because: (3*2+1) in binary is 111 and (3*2+1)*19 = 133 in binary is 10000101, both have 3 bits set to 1.

Pontus von Brömssen, Antidiagonals n = 1..100, flattened

Cf. A000120, A340351, A340069.

Cf. A263132 (superset of 1st row), A007583 (1st row), A299960 (2nd row).

If 2*n = 2^j, then T(n, m) = (1+2^(j+2*j*m))/(2*n+1) for m > 0. In particular:
  T(1, m) = (1+2^(1+2*m))/3 = A007583(m),
  T(2, m) = (1+2^(2+4*m))/5 = A299960(m),
  T(4, m) = (1+2^(3+6*m))/9.
The third row consists of all numbers of the form (1+2^(1+b*3)+2^(2+c*3))/7, where b and c are natural numbers >= 0 and b+c > 0.
The seventh row consists of all numbers of the form (1+2^(1+b*2)+2^(2+c*2)+2^(3+d*2))/15 where b, c, and d are natural numbers >= 0 and b+c+d > 1.

More terms from Pontus von Brömssen, Jan 08 2021

A340349 a(n) is the smallest k such that A292849(k) = 2n-1.

Original entry on oeis.org

1, 3, 13, 5, 57, 35, 21, 9, 241, 219, 49, 45, 169, 83, 73, 17, 993, 59, 941, 53, 3197, 51, 185, 93, 209, 81, 349, 85, 41, 89, 105, 33, 4033, 491, 4749, 247, 449, 227, 429, 363, 3249, 401, 193, 259, 233, 107, 117, 189, 697249, 1355, 173, 517, 473, 1091, 101, 231, 725, 305

Offset: 1

Thomas Scheuerle, Jan 05 2021

This implies that a(n)=k is a solution to A000120(2*n-1) = A000120((2*n-1)*k), where A000120 is the Hamming weight. If no such number exists we define a(n) = 2.
Does this sequence consist only of odd numbers? It appears so.
This will be the case if A292849 contains all odd numbers, because a(n) will then never become 2.
A292849 must contain all odd numbers if two conditions are satisfied:
First: For every odd number 2n-1 there must be an odd k > 1 that satisfies A000120(2n-1) = A000120((2n-1)*k). To prove that this condition is satisfied, it may be helpful that if k*m = 2^j+r, we know that A000120(k*m) = A063787(r) and for each Hamming weight there exists an r such that A063787(r) = A063787(r+1). This allows us to choose r such that the Hamming weight becomes A000120(m) = A063787(r). For a given r, k*m = 2^j+r may have no solutions if k or m are divisors of r, but there may still exist a solution for k*m = 2^j+r+1. Of course this is not a complete proof.
Second: For every n there needs to be a number k such that 2n-1 is the smallest solution to A000120(2n-1) = A000120((2n-1)*k). This is satisfied if no row in A340441 is a subset of a previous row.
No odd number can appear more than once in this sequence, but not all odd numbers will appear, so this sequence is not a permutation of the odd numbers.
This sequence can be constructed from A340441. Start with a(1) = 1, then a(n) is the least number in row n-1 of A340441 that has not already appeared in A340441.

Cf. A000120, A292849, A340069, A263132, A077459, A295827, A340441, A340351 table of multiplications.

function a = A340349(maxA292849)
c = A340351(maxA292849,1);
n = 1; run = 1;
while run == 1
     i = find(c==(n*2)-1);
     if ~isempty(i);
         a(n) = i(1);
         n = n+1;
     else
         run = 0;
     end
end
end
function a = A340351(max_n,max_m)
    for n = 1:max_n
        m = 1; k = 1;
        while m < max_m+1
            c = length(find(bitget(k,1:32)== 1));
            if c == length(find(bitget(n*k,1:32)== 1))
                a(n,m) = k;
                m = m+1;
            end
            k = k +1;
        end
    end
end

f(n) = my(k=1); while ((hammingweight(k)) != hammingweight(k*n), k++); k; \\ A292849
a(n) = my(k=1); while(f(k) != 2*n-1, k++); k; \\ Michel Marcus, Jan 09 2021

A340547 Square array, read by ascending antidiagonals, where row n gives all solutions n > 0 to A000120(n+1) = A000120((n+1)*k), A000120 is the Hamming weight.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 8, 1, 2, 4, 4, 16, 1, 2, 4, 8, 6, 32, 1, 2, 3, 8, 16, 8, 64, 1, 2, 3, 4, 13, 32, 11, 128, 1, 2, 4, 4, 6, 16, 64, 12, 256, 1, 2, 2, 8, 5, 8, 26, 128, 16, 512, 1, 2, 4, 8, 16, 6, 11, 32, 256, 22, 1024

Offset: 1

Thomas Scheuerle, Jan 11 2021

Solutions to related equation A000120(k) = A000120(k*n) are A340351.

The same sequence without leading ones and only odd solutions is A340441.

			Eight initial terms of rows 1-8 are listed below:
   1: 1, 2, 4, 8, 16, 32, 64, 128, ...
   2: 1, 2, 3, 4,  6,  8, 11,  12, ...
   3: 1, 2, 4, 8, 16, 32, 64, 128, ...
   4: 1, 2, 4, 8, 13, 16, 26,  32, ...
   5: 1, 2, 3, 4,  6,  8, 11,  12, ...
   6: 1, 2, 3, 4,  5,  6,  7,   8, ...
   7: 1, 2, 4, 8, 16, 32, 64, 128, ...
   8: 1, 2, 4, 8, 16, 32, 57,  64, ...
T(3,4) = 8 because: (3+1) in binary is 100 and (3*1)*8 = 32 in binary is 100000, both have 1 bit set to 1.

Cf. A000120, A340351, A340069, A340441.

Cf. A263132 (superset of 1st row), A007583 (1st row), A299960 (2nd row).

T(2n, ...)   = 2^{0,1,2,...}, 2^{0,1,2,...} * row n of A340441.
T(4n+1, ...) = 2^{0,1,2,...}, 2^{0,1,2,...} * row n of A340441.
T(2^n, ...)  = 2^{0,1,2,...}.

cp's OEIS Frontend

A340100 Fixed points of A340069.

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A340351 Square array, read by descending antidiagonals, where row n gives all solutions k > 0 to A000120(k)=A000120(k*n), A000120 is the Hamming weight.

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MATLAB

A340441 Square array, read by ascending antidiagonals, where row n gives all odd solutions k > 1 and n > 0 to A000120(2*n+1) = A000120((2*n+1)*k), A000120 is the Hamming weight.

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A340349 a(n) is the smallest k such that A292849(k) = 2n-1.

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PARI

A340547 Square array, read by ascending antidiagonals, where row n gives all solutions n > 0 to A000120(n+1) = A000120((n+1)*k), A000120 is the Hamming weight.

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A340441 Square array, read by ascending antidiagonals, where row n gives all odd solutions k > 1 and n > 0 to A000120(2n+1) = A000120((2n+1)*k), A000120 is the Hamming weight.