cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A368533 Numbers whose binary indices are all squarefree.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119, 512
Offset: 1

Views

Author

Gus Wiseman, Mar 23 2024

Keywords

Comments

The complement first differs from A115419 in having 128.
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.

Examples

			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}
   34:  100010 ~ {2,6}
   35:  100011 ~ {1,2,6}
   36:  100100 ~ {3,6}
   37:  100101 ~ {1,3,6}
   38:  100110 ~ {2,3,6}

Crossrefs

Set multipartitions: A049311, A050320, A089259, A116540.
For prime indices instead of binary indices we have A302478.
The case of prime binary indices is A326782.
The case of squarefree product is A371289.
For prime-power product we have A371290.
For nonprime binary indices we have A371443, composite A371444.
The semiprime case is A371453, squarefree case of A371454.
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. A000040, A001222, A087086, A296119, A326031, A367905, A368109, A371450.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],And@@SquareFreeQ/@bpe[#]&]
  • Python
    from math import isqrt
from sympy import mobius
def A368533(n):
    def f(x,n): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def A005117(n):
        m, k = n, f(n,n)
        while m != k: m, k = k, f(k,n)
        return m
    return sum(1<<A005117(i)-1 for i, j in enumerate(bin(n-1)[:1:-1],1) if j=='1') # Chai Wah Wu, Oct 24 2024

A371289 Numbers whose binary indices have squarefree product.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Mar 25 2024

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

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

A371449 Numbers whose prime indices are not powers of 2.

Original entry on oeis.org

1, 5, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 211, 215
Offset: 1

Views

Author

Gus Wiseman, Mar 31 2024

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
     1: {}        85: {3,7}      169: {6,6}     253: {5,9}
     5: {3}       89: {24}       173: {40}      257: {55}
    11: {5}       97: {25}       179: {41}      263: {56}
    13: {6}      101: {26}       181: {42}      269: {57}
    17: {7}      103: {27}       185: {3,12}    271: {58}
    23: {9}      107: {28}       187: {5,7}     275: {3,3,5}
    25: {3,3}    109: {29}       191: {43}      277: {59}
    29: {10}     113: {30}       193: {44}      281: {60}
    31: {11}     115: {3,9}      197: {45}      283: {61}
    37: {12}     121: {5,5}      199: {46}      289: {7,7}
    41: {13}     125: {3,3,3}    205: {3,13}    293: {62}
    43: {14}     127: {31}       211: {47}      295: {3,17}
    47: {15}     137: {33}       215: {3,14}    299: {6,9}
    55: {3,5}    139: {34}       221: {6,7}     305: {3,18}
    59: {17}     143: {5,6}      223: {48}      307: {63}
    61: {18}     145: {3,10}     227: {49}      313: {65}
    65: {3,6}    149: {35}       229: {50}      317: {66}
    67: {19}     151: {36}       233: {51}      319: {5,10}
    71: {20}     155: {3,11}     235: {3,15}    325: {3,3,6}
    73: {21}     157: {37}       239: {52}      331: {67}
    79: {22}     163: {38}       241: {53}      335: {3,19}
    83: {23}     167: {39}       251: {54}      337: {68}

Crossrefs

Partitions of this type are counted by A101417.
For binary indices instead of prime indices we have A326781.
Requiring powers of two gives A318400, for binary indices A253317.
An opposite version is A371443.
For primes instead of powers of 2 we have A320628.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A057716 lists non-powers of 2.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
Cf. A005117, A326782, A371451, A371444.

Programs

  • Mathematica
    Select[Range[100],And@@Not/@IntegerQ/@Log[2, PrimePi/@First/@FactorInteger[#]]&]

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

Views

Author

Gus Wiseman, Mar 27 2024

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]
Showing 1-4 of 4 results.

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