A352884 -id:A352884 - cp's OEIS frontend

A352575 A109812(n) in binary.

1, 10, 100, 11, 1000, 101, 1010, 10000, 110, 1001, 10010, 1100, 10001, 1110, 100000, 111, 11000, 100001, 10100, 1011, 100100, 10011, 101000, 10101, 100010, 1101, 110000, 1111, 1000000, 10110, 101001, 1000010, 11001, 100110, 1000001, 11010, 100101, 1001000, 10111, 1100000, 11011, 1000100, 100011, 11100, 1000011, 101100, 1010000, 100111, 1011000, 10000000

Offset: 1

N. J. A. Sloane, Apr 04 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Bitmap of a(n), n = 1..2^14, horizontally exaggerating each row by 256X.
N. J. A. Sloane, Right-justified table of n, a(n) for n = 1..20000 [Note this is not a b-file.]
N. J. A. Sloane, Right-justified table of n, a(n) for n = 1..10^6 [gzipped file, copied from A352575] [This file was corrupted, but today I replaced it with a corrected version. - _N. J. A. Sloane_, Apr 11 2022]

Cf. A109812, A352576, A352884.

nn = 50, c[] = 0; a[1] = c[1] = 1; u = 2; Do[If[a[i - 1] == u, While[c[u] > 0, u++]]; k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; Set[{a[i], c[k]}, {k, i}], {i, Length[s] + 1, nn}]; Array[FromDigits@ IntegerDigits[a[#], 2] &, nn] (* _Michael De Vlieger, Apr 05 2022 *)

A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 1, 2, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 5, 3, 5, 3, 6, 3, 5, 2, 7, 1, 5, 6, 3, 5, 5, 3, 5, 6, 3, 8, 2, 9, 3, 5, 4, 5, 5, 3, 8, 5, 1, 8, 5, 3, 7, 5, 3, 10, 6, 3, 6, 6, 5, 6, 5, 6, 5, 5, 4, 8, 3, 9, 5, 5, 7, 2, 11, 3, 10, 3, 10, 6, 6, 9, 5, 8, 6, 5, 8, 5, 10, 6, 12, 6, 10, 6, 5, 10, 4, 6, 8, 5, 13

Offset: 1

Antti Karttunen, Apr 06 2022

Restricted growth sequence transform of A278222(A109812(n)), or equally of, A278222(A351965(n)).

For all i, j: A351578(i) = A351578(j) => a(i) = a(j) => A352884(i) = A352884(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A005940, A046523, A109812, A278222, A286622, A351965, A352884.

Cf. also A351578, A352888.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v351963 = rgs_transform(vector(up_to, n, A046523(A005940(1+A109812(n)))));
A351963(n) = v351963[n];

A351965 The odd part of A109812(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 5, 1, 3, 9, 9, 3, 17, 7, 1, 7, 3, 33, 5, 11, 9, 19, 5, 21, 17, 13, 3, 15, 1, 11, 41, 33, 25, 19, 65, 13, 37, 9, 23, 3, 27, 17, 35, 7, 67, 11, 5, 39, 11, 1, 29, 49, 129, 15, 97, 65, 45, 41, 33, 21, 69, 25, 73, 13, 37, 49, 35, 7, 71, 17, 51, 19, 131, 15, 3, 31, 5, 75, 9, 43, 21, 137, 27, 193, 23, 81

Offset: 1

Antti Karttunen, Apr 06 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A000265, A109812, A351963, A351964, A352884 (binary weight).

v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A000265(n) = (n>>valuation(n,2));
A351965(n) = A000265(A109812(n));

a(n) = A000265(A109812(n)).

a(n) = A109812(n) / (2^A351964(n)).

A352889 Number of runs in the binary expansion of A109812(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 07 2022

			   n   A109812(n)  [in base-2]   a(n) = number of runs
-----+-------------------------------------------------
   1 |          1       [1],      1
   2 |          2      [10],      2
   3 |          4     [100],      2
   4 |          3      [11],      1
   5 |          8    [1000],      2
   6 |          5     [101],      3
   7 |         10    [1010],      4
   8 |         16   [10000],      2
   9 |          6     [110],      2
  10 |          9    [1001],      3
  11 |         18   [10010],      4

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A005811, A109812, A352575, A352884, A352888.

v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A352889(n) = A005811(A109812(n));

a(n) = A005811(A109812(n)).

A352791 a(n) is the number of numbers k < n such that A109812(k) AND A109812(n) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 3, 7, 3, 4, 5, 5, 6, 3, 14, 3, 8, 10, 8, 4, 11, 5, 12, 5, 11, 5, 14, 3, 28, 7, 8, 18, 8, 8, 18, 8, 9, 21, 5, 26, 5, 21, 9, 11, 11, 11, 25, 6, 15, 49, 7, 16, 29, 8, 16, 31, 8, 17, 33, 14, 14, 15, 16, 16, 16, 15, 15, 18, 8, 40, 9, 18, 21, 11

Offset: 1

Rémy Sigrist, Apr 03 2022

The magnitude of a(n) is related to A352884(n), the Hamming weight of A109812(n) (see illustration in Links section).

			The first terms, alongside the binary expansion of A109812(n) and the corresponding k's, are:
  n   a(n)  bin(b(n))  k's
  --  ----  ---------  -----------------------------------------------
   1     0          1  []
   2     1         10  [1]
   3     2        100  [1, 2]
   4     1         11  [3]
   5     4       1000  [1, 2, 3, 4]
   6     2        101  [2, 5]
   7     3       1010  [1, 3, 6]
   8     7      10000  [1, 2, 3, 4, 5, 6, 7]
   9     3        110  [1, 5, 8]
  10     4       1001  [2, 3, 8, 9]
  11     5      10010  [1, 3, 5, 6, 10]
  12     5       1100  [1, 2, 4, 8, 11]
  13     6      10001  [2, 3, 5, 7, 9, 12]
  14     3       1110  [1, 8, 13]
  15    14     100000  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
  16     3        111  [5, 8, 15]

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of the Hamming weight of A109812(n))
Rémy Sigrist, C++ program

Cf. A109812, A352884.

a(n) <= n-1 with equality iff A109812(n) is a power of 2.

cp's OEIS Frontend

A352575 A109812(n) in binary.

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A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

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PARI

A351965 The odd part of A109812(n).

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A352889 Number of runs in the binary expansion of A109812(n).

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A352791 a(n) is the number of numbers k < n such that A109812(k) AND A109812(n) = 0 (where AND denotes the bitwise AND operator).

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