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 1-5 of 5 results.

A372429 Sum of binary indices of prime(n). Sum of positions of ones in the reversed binary expansion of prime(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 6, 8, 11, 13, 15, 10, 11, 13, 16, 15, 18, 19, 10, 13, 12, 17, 15, 17, 14, 17, 19, 20, 21, 19, 28, 11, 13, 15, 17, 19, 21, 17, 20, 22, 22, 23, 29, 16, 19, 21, 23, 30, 24, 25, 26, 31, 27, 33, 10, 15, 17, 19, 18, 19, 21, 19, 23, 26, 25, 28, 23
Offset: 1

Views

Author

Gus Wiseman, May 02 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.
Do 2, 3, 4, 7, 12, 14 appear just once?
Are 1, 5, 9 missing?
The above questions hold true up to n = 10^6. - John Tyler Rascoe, May 21 2024

Examples

			The primes together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   5:     101 ~ {1,3}
   7:     111 ~ {1,2,3}
  11:    1011 ~ {1,2,4}
  13:    1101 ~ {1,3,4}
  17:   10001 ~ {1,5}
  19:   10011 ~ {1,2,5}
  23:   10111 ~ {1,2,3,5}
  29:   11101 ~ {1,3,4,5}
  31:   11111 ~ {1,2,3,4,5}
  37:  100101 ~ {1,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  47:  101111 ~ {1,2,3,4,6}
  53:  110101 ~ {1,3,5,6}
  59:  111011 ~ {1,2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}
  67: 1000011 ~ {1,2,7}
  71: 1000111 ~ {1,2,3,7}
  73: 1001001 ~ {1,4,7}
  79: 1001111 ~ {1,2,3,4,7}

Links

Crossrefs

The number instead of sum of binary indices is A014499.
Restriction of A029931 (sum of binary indices) to the primes A000040.
The maximum instead of sum of binary indices is A035100, see also A023506.
Row-sums of A372471.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020.
A056239 adds up prime indices.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A372427 lists numbers whose binary and prime indices have the same sum.
Cf. A005940, A030101, A059893, A071814, A230877, A231204, A358134, A359359, A359401, A372430-A372437, A372441.

Programs

  • Mathematica
    bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[bix[Prime[n]]],{n,100}]

Formula

a(n) = A029931(prime(n)).

A372432 Positive integers k such that the prime indices of k are not disjoint from the binary indices of k.

Original entry on oeis.org

3, 5, 6, 14, 15, 18, 20, 22, 27, 28, 30, 39, 42, 45, 51, 52, 54, 55, 56, 60, 63, 66, 68, 70, 75, 77, 78, 85, 87, 88, 90, 91, 95, 99, 100, 102, 104, 105, 110, 111, 114, 117, 119, 121, 123, 125, 126, 133, 135, 138, 140, 147, 150, 152, 154, 159, 162, 165, 168
Offset: 1

Views

Author

Gus Wiseman, May 03 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.
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 binary indices of 18 are {2,5}, and the prime indices are {1,2,2}, so 18 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
The terms together with their binary expansions and binary indices begin:
    3:      11 ~ {1,2}
    5:     101 ~ {1,3}
    6:     110 ~ {2,3}
   14:    1110 ~ {2,3,4}
   15:    1111 ~ {1,2,3,4}
   18:   10010 ~ {2,5}
   20:   10100 ~ {3,5}
   22:   10110 ~ {2,3,5}
   27:   11011 ~ {1,2,4,5}
   28:   11100 ~ {3,4,5}
   30:   11110 ~ {2,3,4,5}

Crossrefs

For subset instead of overlap we have A372430.
The complement is A372431.
Equal lengths: A071814, zeros of A372441.
Equal sums: A372427, zeros of A372428.
Equal maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

Programs

  • Mathematica
    bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[bix[#],prix[#]]!={}&]

A372439 Numbers k such that the least binary index of k plus the least prime index of k is odd.

Original entry on oeis.org

2, 3, 6, 7, 8, 9, 10, 13, 14, 15, 18, 19, 21, 22, 24, 26, 27, 29, 30, 32, 33, 34, 37, 38, 39, 40, 42, 43, 45, 46, 49, 50, 51, 53, 54, 56, 57, 58, 61, 62, 63, 66, 69, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 86, 87, 88, 89, 90, 91, 93, 94, 96, 98, 99, 101, 102
Offset: 1

Views

Author

Gus Wiseman, May 06 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.
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 (center), their binary indices (left), and their prime indices (right) begin:
        {2}   2  (1)
      {1,2}   3  (2)
      {2,3}   6  (2,1)
    {1,2,3}   7  (4)
        {4}   8  (1,1,1)
      {1,4}   9  (2,2)
      {2,4}  10  (3,1)
    {1,3,4}  13  (6)
    {2,3,4}  14  (4,1)
  {1,2,3,4}  15  (3,2)
      {2,5}  18  (2,2,1)
    {1,2,5}  19  (8)
    {1,3,5}  21  (4,2)
    {2,3,5}  22  (5,1)
      {4,5}  24  (2,1,1,1)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
  {1,3,4,5}  29  (10)
  {2,3,4,5}  30  (3,2,1)
        {6}  32  (1,1,1,1,1)
      {1,6}  33  (5,2)
      {2,6}  34  (7,1)

Crossrefs

Positions of odd terms in A372437.
The complement is 1 followed by A372440.
For sum (A372428, zeros A372427) we have A372586, complement A372587.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
For length (A372441, zeros A071814) we have A372590, complement A372591.
A003963 gives product of prime indices, binary A096111.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A061712, A174090, A243055, A359495, A372429, A372430, A372431, A372432, A372438, A372471.

Programs

  • Mathematica
    bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[Min[bix[#]]+Min[prix[#]]]&]

A372440 Numbers k such that the least binary index of k plus the least prime index of k is even.

Original entry on oeis.org

4, 5, 11, 12, 16, 17, 20, 23, 25, 28, 31, 35, 36, 41, 44, 47, 48, 52, 55, 59, 60, 64, 65, 67, 68, 73, 76, 80, 83, 84, 85, 92, 95, 97, 100, 103, 108, 109, 112, 115, 116, 121, 124, 125, 127, 132, 137, 140, 143, 144, 145, 148, 149, 155, 156, 157, 164, 167, 172
Offset: 1

Views

Author

Gus Wiseman, May 06 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.
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 (center), their binary indices (left), and their prime indices (right) begin:
          {3}   4  (1,1)
        {1,3}   5  (3)
      {1,2,4}  11  (5)
        {3,4}  12  (2,1,1)
          {5}  16  (1,1,1,1)
        {1,5}  17  (7)
        {3,5}  20  (3,1,1)
    {1,2,3,5}  23  (9)
      {1,4,5}  25  (3,3)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
      {1,2,6}  35  (4,3)
        {3,6}  36  (2,2,1,1)
      {1,4,6}  41  (13)
      {3,4,6}  44  (5,1,1)
  {1,2,3,4,6}  47  (15)
        {5,6}  48  (2,1,1,1,1)
      {3,5,6}  52  (6,1,1)
  {1,2,3,5,6}  55  (5,3)
  {1,2,4,5,6}  59  (17)
    {3,4,5,6}  60  (3,2,1,1)
          {7}  64  (1,1,1,1,1,1)

Crossrefs

For sum (A372428, zeros A372427) we have A372587, complement A372586.
Positions of even terms in A372437.
The complement is 1 followed by A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
A003963 gives product of prime indices, binary A096111.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A061712, A174090, A243055, A359402, A359495, A372429, A372430, A372431, A372432, A372438, A372471.

Programs

  • Mathematica
    bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Min[bix[#]]+Min[prix[#]]]&]

A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 26, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 53, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 84, 86, 89, 92, 93, 94, 96, 97, 98, 101
Offset: 1

Views

Author

Gus Wiseman, May 03 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.
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 binary indices of 65 are {1,7}, and the prime indices are {3,6}, so 65 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    13: {6}
    16: {1,1,1,1}
The terms together with their binary expansions and binary indices begin:
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   7:     111 ~ {1,2,3}
   8:    1000 ~ {4}
   9:    1001 ~ {1,4}
  10:    1010 ~ {2,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  13:    1101 ~ {1,3,4}
  16:   10000 ~ {5}

Crossrefs

For subset instead of disjoint we have A372430.
The complement is A372432.
Equal lengths: A071814, zeros of A372441.
Equal sums: A372427, zeros of A372428.
Equal maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

Programs

  • Mathematica
    bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[bix[#],prix[#]]=={}&]
Showing 1-5 of 5 results.

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