A253317 -id:A253317 - cp's OEIS frontend

A355340 a(0) = 0; for n >= 1, a(n) = a(n-1) XOR A001511(n), where XOR denotes bitwise exclusive-or (A003987) and A001511 is the binary ruler function.

0, 1, 3, 2, 1, 0, 2, 3, 7, 6, 4, 5, 6, 7, 5, 4, 1, 0, 2, 3, 0, 1, 3, 2, 6, 7, 5, 4, 7, 6, 4, 5, 3, 2, 0, 1, 2, 3, 1, 0, 4, 5, 7, 6, 5, 4, 6, 7, 2, 3, 1, 0, 3, 2, 0, 1, 5, 4, 6, 7, 4, 5, 7, 6, 1, 0, 2, 3, 0, 1, 3, 2, 6, 7, 5, 4, 7, 6, 4, 5, 0, 1, 3, 2, 1, 0, 2, 3, 7, 6, 4, 5, 6, 7, 5, 4, 2, 3, 1, 0, 3, 2, 0, 1, 5

Offset: 0

Peter Munn, Jun 29 2022

Related to the Thue-Morse sequence, A010060, which gives the rightmost binary bit of each term. The next bit is given by the closely related A269723.
If we replace A001511(n) in the definition by A006519(n) = 2^(A001511(n)-1) we get Gray code (A003188).
Interesting symmetries of the sequence seem more apparent with the terms aligned in suitable periods, such as the arrangement in the example section.

			Initial terms arranged in periods of 16, with deliberate periodic spacing:
  0,1,3,2,  1,0,2,3,     7,6,4,5,  6,7,5,4,
  1,0,2,3,  0,1,3,2,     6,7,5,4,  7,6,4,5,
  3,2,0,1,  2,3,1,0,     4,5,7,6,  5,4,6,7,
  2,3,1,0,  3,2,0,1,     5,4,6,7,  4,5,7,6,
.
  1,0,2,3,  0,1,3,2,     6,7,5,4,  7,6,4,5,
  0,1,3,2,  1,0,2,3,     7,6,4,5,  6,7,5,4,
  2,3,1,0,  3,2,0,1,     5,4,6,7,  4,5,7,6,
  3,2,0,1,  2,3,1,0,     4,5,7,6,  5,4,6,7,
...
Note that when the arrangement is partitioned regularly into 2 X 2, 4 X 4 or 8 X 8 squares, the terms on any diagonal of a square share the same value. Note also the symmetry of the terms on the squares' circumferences.

Paolo Xausa, Table of n, a(n) for n = 0..16383
Rémy Sigrist, Colored representation of the first 2^15 terms as 128 rows of 256 terms (the color is function of a(x + 256*y), x = 0..255, y = 0..127)
Index entries for sequences related to binary expansion of n

Comparable sequences: A010060, A261283, A269723.
Cf. A001511, A003188, A003987, A006519.
Positions of: odd numbers: A000069, even numbers: A001969, previously unseen numbers: A253317 (apparently).

Block[{k = 0}, NestList[BitXor[#, IntegerExponent[k += 2, 2]] &, 0, 100]] (* Paolo Xausa, May 29 2024 *)

A010060(n) = a(n) mod 2.

A269723(n) = floor(a(n)/2) mod 2.

A358126 Replace 2^k in binary expansion of n with 2^(2^k).

Original entry on oeis.org

0, 2, 4, 6, 16, 18, 20, 22, 256, 258, 260, 262, 272, 274, 276, 278, 65536, 65538, 65540, 65542, 65552, 65554, 65556, 65558, 65792, 65794, 65796, 65798, 65808, 65810, 65812, 65814, 4294967296, 4294967298, 4294967300

Offset: 0

Tilman Piesk, Oct 30 2022

Sums of distinct terms of A001146.
The name "ballooned integers" is proposed for this sequence.
a(n) is the index of the first occurrence of n in A253315.

			Let    n   =     25  =  1 +   8 +    16  =     2^0  +    2^3  +    2^4.
Then a(n)  =  65794  =  2 + 256 + 65536  =  2^(2^0) + 2^(2^3) + 2^(2^4).
The binary indices of n are {0, 3, 4}. Those of a(n) are {1, 8, 16}.

Tilman Piesk, Table of n, a(n) for n = 0..4095
Tilman Piesk, Boolean Walsh functions and corresponding balloons
Tilman Piesk, First 32 entries in little-endian binary

Cf. A253317, A001146, A060803, A133457, A197819, A228539, A253315.

a := proc(n) select(d -> d[2] <> 0, ListTools:-Enumerate(convert(n,base,2))):
add(2^(2^(%[j][1] - 1)), j = 1..nops(%)) end: seq(a(n), n = 0..34); # Peter Luschny, Oct 31 2022

a[n_] := Total[2^(2^Range[If[n == 0, 1, IntegerLength[n,2]] - 1, 0, -1]) * IntegerDigits[n, 2]]; Array[a, 35, 0] (* Amiram Eldar, Oct 31 2022 *)

a(n) = my(d=Vecrev(digits(n,2))); for (k=1, #d, d[k] *= 2^(2^(k-1))); vecsum(d); \\ Michel Marcus, Oct 31 2022

def a(n):
    binary_string = "{0:b}".format(n)[::-1]  # little-endian
    result = 0
    for i, binary_digit in enumerate(binary_string):
        if binary_digit == '1':
            result += 1 << (1 << i)  # 2 ** (2 ** i)
    return result

If n = Sum_{i=0..k} 2^s_i, then a(n) = Sum_{i=0..k} 2^(2^s_i).
a(n) = 2 * A253317(n+1).
a(2^n-1) = A060803(n-1) for n >= 1.
a(2^n) = A001146(n).
A197819[m, a(n)] = A228539[m, n]. (Compare link about Boolean Walsh functions.)

A368531 Numbers whose binary indices are all powers of 3, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

Original entry on oeis.org

0, 1, 4, 5, 256, 257, 260, 261, 67108864, 67108865, 67108868, 67108869, 67109120, 67109121, 67109124, 67109125, 1208925819614629174706176, 1208925819614629174706177, 1208925819614629174706180, 1208925819614629174706181, 1208925819614629174706432

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For powers of 2 instead of 3 we have A253317.

			The terms together with their binary expansions and binary indices begin:
         0:                           0 ~ {}
         1:                           1 ~ {1}
         4:                         100 ~ {3}
         5:                         101 ~ {1,3}
       256:                   100000000 ~ {9}
       257:                   100000001 ~ {1,9}
       260:                   100000100 ~ {3,9}
       261:                   100000101 ~ {1,3,9}
  67108864: 100000000000000000000000000 ~ {27}
  67108865: 100000000000000000000000001 ~ {1,27}
  67108868: 100000000000000000000000100 ~ {3,27}
  67108869: 100000000000000000000000101 ~ {1,3,27}
  67109120: 100000000000000000100000000 ~ {9,27}
  67109121: 100000000000000000100000001 ~ {1,9,27}
  67109124: 100000000000000000100000100 ~ {3,9,27}
  67109125: 100000000000000000100000101 ~ {1,3,9,27}

Michael De Vlieger, Table of n, a(n) for n = 1..256

A000244 lists powers of 3.
A048793 lists binary indices, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A058891, A062050, A072639, A253317, A326031, A326675, A326702, A367912, A368183, A368109.

Select[Range[0,10000],IntegerQ[Log[3,Times@@Join@@Position[Reverse[IntegerDigits[#,2]],1]]]&]
(* Second program *)
{0}~Join~Array[FromDigits[Reverse@ ReplacePart[ConstantArray[0, Max[#]], Map[# -> 1 &, #]], 2] &[3^(Position[Reverse@ IntegerDigits[#, 2], 1][[;; , 1]] - 1)] &, 255] (* Michael De Vlieger, Dec 29 2023 *)

a(3^n) = 2^(3^n - 1).

A371290 Numbers whose product of binary indices is a prime power > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 64, 65, 128, 129, 130, 131, 136, 137, 138, 139, 256, 257, 260, 261, 1024, 1025, 4096, 4097, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 65536, 65537, 262144

Offset: 1

Gus Wiseman, Mar 27 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
       1:                   1 ~ {1}
       2:                  10 ~ {2}
       3:                  11 ~ {1,2}
       4:                 100 ~ {3}
       5:                 101 ~ {1,3}
       8:                1000 ~ {4}
       9:                1001 ~ {1,4}
      10:                1010 ~ {2,4}
      11:                1011 ~ {1,2,4}
      16:               10000 ~ {5}
      17:               10001 ~ {1,5}
      64:             1000000 ~ {7}
      65:             1000001 ~ {1,7}
     128:            10000000 ~ {8}
     129:            10000001 ~ {1,8}
     130:            10000010 ~ {2,8}
     131:            10000011 ~ {1,2,8}
     136:            10001000 ~ {4,8}
     137:            10001001 ~ {1,4,8}
     138:            10001010 ~ {2,4,8}
     139:            10001011 ~ {1,2,4,8}
     256:           100000000 ~ {9}
     257:           100000001 ~ {1,9}
     260:           100000100 ~ {3,9}
     261:           100000101 ~ {1,3,9}
    1024:         10000000000 ~ {11}
    1025:         10000000001 ~ {1,11}
    4096:       1000000000000 ~ {13}
    4097:       1000000000001 ~ {1,13}
   32768:    1000000000000000 ~ {16}

For powers of 2 we have A253317.
For prime indices we have A320698.
For squarefree numbers instead of prime powers we have A371289.
A000040 lists prime numbers.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A005117, A326782, A368533, A371292, A371443, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]

cp's OEIS Frontend

A355340 a(0) = 0; for n >= 1, a(n) = a(n-1) XOR A001511(n), where XOR denotes bitwise exclusive-or (A003987) and A001511 is the binary ruler function.

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A358126 Replace 2^k in binary expansion of n with 2^(2^k).

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A368531 Numbers whose binary indices are all powers of 3, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

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A371290 Numbers whose product of binary indices is a prime power > 1.

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