A371449 -id:A371449 - cp's OEIS frontend

A371289 Numbers whose binary indices have squarefree product.

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 48, 49, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 112, 113, 512, 513, 516, 517, 576, 577, 580, 581, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1040, 1041, 1042

Offset: 1

Gus Wiseman, Mar 25 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:
     0:              0 ~ {}
     1:              1 ~ {1}
     2:             10 ~ {2}
     3:             11 ~ {1,2}
     4:            100 ~ {3}
     5:            101 ~ {1,3}
     6:            110 ~ {2,3}
     7:            111 ~ {1,2,3}
    16:          10000 ~ {5}
    17:          10001 ~ {1,5}
    18:          10010 ~ {2,5}
    19:          10011 ~ {1,2,5}
    20:          10100 ~ {3,5}
    21:          10101 ~ {1,3,5}
    22:          10110 ~ {2,3,5}
    23:          10111 ~ {1,2,3,5}
    32:         100000 ~ {6}
    33:         100001 ~ {1,6}
    48:         110000 ~ {5,6}
    49:         110001 ~ {1,5,6}
    64:        1000000 ~ {7}
    65:        1000001 ~ {1,7}
    66:        1000010 ~ {2,7}

For prime instead of binary indices we have A302505.
For squarefree parts we have A368533, for prime indices A302478.
A005117 lists squarefree numbers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A325118, A326782, A371290, A371291, A371292, A371293, A371443, A371446, A371448, A371449, A371452, A371453.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SquareFreeQ[Times@@bpe[#]]&]

A371443 Numbers whose binary indices are nonprime numbers.

Original entry on oeis.org

1, 8, 9, 32, 33, 40, 41, 128, 129, 136, 137, 160, 161, 168, 169, 256, 257, 264, 265, 288, 289, 296, 297, 384, 385, 392, 393, 416, 417, 424, 425, 512, 513, 520, 521, 544, 545, 552, 553, 640, 641, 648, 649, 672, 673, 680, 681, 768, 769, 776, 777, 800, 801, 808

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

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}
    8:       1000 ~ {4}
    9:       1001 ~ {1,4}
   32:     100000 ~ {6}
   33:     100001 ~ {1,6}
   40:     101000 ~ {4,6}
   41:     101001 ~ {1,4,6}
  128:   10000000 ~ {8}
  129:   10000001 ~ {1,8}
  136:   10001000 ~ {4,8}
  137:   10001001 ~ {1,4,8}
  160:   10100000 ~ {6,8}
  161:   10100001 ~ {1,6,8}
  168:   10101000 ~ {4,6,8}
  169:   10101001 ~ {1,4,6,8}
  256:  100000000 ~ {9}
  257:  100000001 ~ {1,9}
  264:  100001000 ~ {4,9}
  265:  100001001 ~ {1,4,9}
  288:  100100000 ~ {6,9}
  289:  100100001 ~ {1,6,9}
  296:  100101000 ~ {4,6,9}

For powers of 2 instead of nonprime numbers we have A253317.
For prime indices instead of binary indices we have A320628.
For prime instead of nonprime we have A326782.
For composite numbers we have A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
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. A001222, A005117, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],And@@Not/@PrimeQ/@bpe[#]&]

A371444 Numbers whose binary indices are composite numbers.

Original entry on oeis.org

8, 32, 40, 128, 136, 160, 168, 256, 264, 288, 296, 384, 392, 416, 424, 512, 520, 544, 552, 640, 648, 672, 680, 768, 776, 800, 808, 896, 904, 928, 936, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2304, 2312, 2336, 2344, 2432, 2440, 2464, 2472, 2560, 2568

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

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:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   128:       10000000 ~ {8}
   136:       10001000 ~ {4,8}
   160:       10100000 ~ {6,8}
   168:       10101000 ~ {4,6,8}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   384:      110000000 ~ {8,9}
   392:      110001000 ~ {4,8,9}
   416:      110100000 ~ {6,8,9}
   424:      110101000 ~ {4,6,8,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   640:     1010000000 ~ {8,10}
   648:     1010001000 ~ {4,8,10}
   672:     1010100000 ~ {6,8,10}

For powers of 2 instead of composite numbers we have A253317.
For prime indices we have the even case of A320628.
For prime instead of composite we have A326782.
This is the even case of A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
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. A001222, A005117, A055887, A320629, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[#]&&And@@Not/@PrimeQ/@bpe[#]&]

cp's OEIS Frontend

A371289 Numbers whose binary indices have squarefree product.

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A371443 Numbers whose binary indices are nonprime numbers.

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A371444 Numbers whose binary indices are composite numbers.

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