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.

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A372438 Least binary index equals greatest prime index.

Original entry on oeis.org

6, 18, 20, 54, 56, 60, 100, 162, 168, 176, 180, 280, 300, 392, 416, 486, 500, 504, 528, 540, 840, 880, 900, 1088, 1176, 1232, 1248, 1400, 1458, 1500, 1512, 1584, 1620, 1936, 1960, 2080, 2432, 2500, 2520, 2640, 2700, 2744, 2912, 3264, 3528, 3696, 3744, 4200
Offset: 1

Views

Author

Gus Wiseman, May 04 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.
Are there any squarefree terms > 6?

Examples

			The binary indices of 60 are {3,4,5,6}, the prime indices are {1,1,2,3}, and 3 = 3, so 60 is in the sequence.
The terms together with their prime indices begin:
     6: {1,2}
    18: {1,2,2}
    20: {1,1,3}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
   100: {1,1,3,3}
   162: {1,2,2,2,2}
   168: {1,1,1,2,4}
   176: {1,1,1,1,5}
   180: {1,1,2,2,3}
   280: {1,1,1,3,4}
   300: {1,1,2,3,3}
The terms together with their binary expansions and binary indices begin:
     6:            110 ~ {2,3}
    18:          10010 ~ {2,5}
    20:          10100 ~ {3,5}
    54:         110110 ~ {2,3,5,6}
    56:         111000 ~ {4,5,6}
    60:         111100 ~ {3,4,5,6}
   100:        1100100 ~ {3,6,7}
   162:       10100010 ~ {2,6,8}
   168:       10101000 ~ {4,6,8}
   176:       10110000 ~ {5,6,8}
   180:       10110100 ~ {3,5,6,8}
   280:      100011000 ~ {4,5,9}
   300:      100101100 ~ {3,4,6,9}

Crossrefs

Same length: A071814, zeros of A372441.
Same sum: A372427, zeros of A372428.
Same 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, A014499, A030101, A061712, A318283, A355536, A359401, A359402, A372429-A372433.

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[1000],Min[bix[#]]==Max[prix[#]]&]

Formula

A001511(a(n)) = A061395(a(n)).

A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 15, 16, 17, 20, 21, 29, 32, 36, 42, 43, 45, 46, 47, 48, 51, 53, 54, 55, 59, 60, 61, 63, 64, 65, 66, 67, 68, 71, 73, 78, 79, 80, 81, 84, 89, 91, 93, 94, 95, 97, 99, 101, 105, 110, 111, 113, 114, 115, 116, 118, 119, 121, 122, 125, 127
Offset: 1

Views

Author

Gus Wiseman, May 14 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.
The even version is A372587.

Examples

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {1}   1  ()
            {2}   2  (1)
          {1,2}   3  (2)
            {3}   4  (1,1)
          {1,3}   5  (3)
            {4}   8  (1,1,1)
          {1,4}   9  (2,2)
          {3,4}  12  (2,1,1)
      {1,2,3,4}  15  (3,2)
            {5}  16  (1,1,1,1)
          {1,5}  17  (7)
          {3,5}  20  (3,1,1)
        {1,3,5}  21  (4,2)
      {1,3,4,5}  29  (10)
            {6}  32  (1,1,1,1,1)
          {3,6}  36  (2,2,1,1)
        {2,4,6}  42  (4,2,1)
      {1,2,4,6}  43  (14)
      {1,3,4,6}  45  (3,2,2)
      {2,3,4,6}  46  (9,1)
    {1,2,3,4,6}  47  (15)
          {5,6}  48  (2,1,1,1,1)

Crossrefs

Positions of odd terms in A372428, zeros A372427.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372587.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
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, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Programs

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

Formula

Numbers k such that A029931(k) + A056239(k) is odd.

A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.

Original entry on oeis.org

3, 4, 5, 9, 12, 13, 14, 16, 17, 20, 22, 23, 25, 30, 31, 35, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 58, 61, 63, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124
Offset: 1

Views

Author

Gus Wiseman, May 14 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.
The odd version is A372588.

Examples

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1,2}   3  (2)
          {3}   4  (1,1)
        {1,3}   5  (3)
        {1,4}   9  (2,2)
        {3,4}  12  (2,1,1)
      {1,3,4}  13  (6)
      {2,3,4}  14  (4,1)
          {5}  16  (1,1,1,1)
        {1,5}  17  (7)
        {3,5}  20  (3,1,1)
      {2,3,5}  22  (5,1)
    {1,2,3,5}  23  (9)
      {1,4,5}  25  (3,3)
    {2,3,4,5}  30  (3,2,1)
  {1,2,3,4,5}  31  (11)
      {1,2,6}  35  (4,3)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
      {2,3,6}  38  (8,1)
    {1,2,3,6}  39  (6,2)
      {2,4,6}  42  (4,2,1)
    {1,2,4,6}  43  (14)

Crossrefs

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
Positions of even terms in A372442, zeros A372436.
The complement is A372588.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum: A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
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, A006141, A066207, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Programs

  • Mathematica
    Select[Range[2,100],EvenQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Formula

Numbers k such that A070939(k) + A061395(k) is even.

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

Original entry on oeis.org

1, 3, 4, 5, 12, 14, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 35, 38, 43, 45, 48, 49, 53, 55, 56, 62, 63, 64, 66, 68, 69, 71, 72, 74, 75, 78, 80, 81, 82, 83, 84, 87, 88, 89, 91, 92, 93, 94, 99, 100, 101, 102, 104, 105, 108, 113, 114, 115, 116, 118, 120
Offset: 1

Views

Author

Gus Wiseman, May 14 2024

Keywords

Comments

The even version is A372591.

Examples

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1}   1  ()
      {1,2}   3  (2)
        {3}   4  (1,1)
      {1,3}   5  (3)
      {3,4}  12  (2,1,1)
    {2,3,4}  14  (4,1)
        {5}  16  (1,1,1,1)
      {1,5}  17  (7)
      {2,5}  18  (2,2,1)
      {3,5}  20  (3,1,1)
    {1,3,5}  21  (4,2)
    {2,3,5}  22  (5,1)
  {1,2,3,5}  23  (9)
    {1,4,5}  25  (3,3)
    {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)
    {1,2,6}  35  (4,3)
    {2,3,6}  38  (8,1)
  {1,2,4,6}  43  (14)
  {1,3,4,6}  45  (3,2,2)

Crossrefs

For sum (A372428, zeros A372427) we have A372586, complement A372587.
For minimum (A372437) we have A372439, complement A372440.
Positions of odd terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372591.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Programs

  • Mathematica
    Select[Range[100],OddQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

Original entry on oeis.org

2, 3, 11, 23, 29, 41, 43, 61, 71, 79, 89, 101, 103, 113, 131, 137, 149, 151, 163, 181, 191, 197, 211, 239, 269, 271, 281, 293, 307, 331, 349, 353, 373, 383, 401, 433, 457, 491, 503, 509, 523, 541, 547, 593, 641, 683, 701, 709, 743, 751, 761, 773, 827, 863, 887
Offset: 1

Views

Author

Gus Wiseman, May 19 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.
The indices of these primes are A372886.

Examples

			The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    2:         10 ~ {2}
    3:         11 ~ {1,2}
   11:       1011 ~ {1,2,4}
   23:      10111 ~ {1,2,3,5}
   29:      11101 ~ {1,3,4,5}
   41:     101001 ~ {1,4,6}
   43:     101011 ~ {1,2,4,6}
   61:     111101 ~ {1,3,4,5,6}
   71:    1000111 ~ {1,2,3,7}
   79:    1001111 ~ {1,2,3,4,7}
   89:    1011001 ~ {1,4,5,7}
  101:    1100101 ~ {1,3,6,7}
  103:    1100111 ~ {1,2,3,6,7}
  113:    1110001 ~ {1,5,6,7}
  131:   10000011 ~ {1,2,8}
  137:   10001001 ~ {1,4,8}
  149:   10010101 ~ {1,3,5,8}
  151:   10010111 ~ {1,2,3,5,8}
  163:   10100011 ~ {1,2,6,8}
  181:   10110101 ~ {1,3,5,6,8}
  191:   10111111 ~ {1,2,3,4,5,6,8}
  197:   11000101 ~ {1,3,7,8}

Links

Crossrefs

For prime instead of binary indices we have A006450, prime case of A316091.
Prime numbers p such that A029931(p) is also prime.
Prime case of A372689.
The indices of these primes are A372886.
A000040 lists the prime numbers, A014499 their binary indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.

Programs

  • Maple
    filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, Jun 19 2025
  • Mathematica
    Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]

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[#]]]&]

A372587 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is even.

Original entry on oeis.org

6, 7, 10, 11, 13, 14, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 44, 49, 50, 52, 56, 57, 58, 62, 69, 70, 72, 74, 75, 76, 77, 82, 83, 85, 86, 87, 88, 90, 92, 96, 98, 100, 102, 103, 104, 106, 107, 108, 109, 112, 117, 120, 123
Offset: 1

Views

Author

Gus Wiseman, May 14 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.
The odd version is A372586.

Examples

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {2,3}   6  (2,1)
          {1,2,3}   7  (4)
            {2,4}  10  (3,1)
          {1,2,4}  11  (5)
          {1,3,4}  13  (6)
          {2,3,4}  14  (4,1)
            {2,5}  18  (2,2,1)
          {1,2,5}  19  (8)
          {2,3,5}  22  (5,1)
        {1,2,3,5}  23  (9)
            {4,5}  24  (2,1,1,1)
          {1,4,5}  25  (3,3)
          {2,4,5}  26  (6,1)
        {1,2,4,5}  27  (2,2,2)
          {3,4,5}  28  (4,1,1)
        {2,3,4,5}  30  (3,2,1)
      {1,2,3,4,5}  31  (11)
            {1,6}  33  (5,2)
            {2,6}  34  (7,1)
          {1,2,6}  35  (4,3)
          {1,3,6}  37  (12)
          {2,3,6}  38  (8,1)

Crossrefs

Positions of even terms in A372428, zeros A372427.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372586.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum: A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
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, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Programs

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

Formula

Numbers k such that A029931(k) + A056239(k) is even.

A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157
Offset: 0

Views

Author

Gus Wiseman, May 15 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.
Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

Examples

			The a(2) = 1 through a(17) = 8 prime numbers:
  2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
              7       13     67   41       71       263  2053  523
                             37   23       43       139  1033  269
                                           29       83   193   163
                                                    53   47    149
                                                    31         101
                                                               89
                                                               79
The a(2) = 1 through a(11) = 3 strict partitions:
  (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
                        (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
                                                      (7,2,1)  (5,3,2,1)

Crossrefs

For all positive integers (not just prime) we get A000009.
Number of prime numbers p with A029931(p) = n.
For odd instead of prime we have A096765, even A025147, non-strict A087787
Number of times n appears in A372429.
Number of rows of A372471 with sum n.
The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).
These (strict) partitions have Heinz numbers A372851.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 lists binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
- reverse A272020
A058698 counts partitions of prime numbers, strict A064688.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A372850 Numbers whose distinct prime indices are the binary indices of some prime number.

Original entry on oeis.org

3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 30, 36, 40, 42, 44, 46, 48, 50, 54, 60, 66, 70, 72, 80, 81, 84, 88, 90, 92, 96, 100, 102, 108, 114, 118, 120, 126, 130, 132, 140, 144, 150, 160, 162, 168, 176, 180, 182, 184, 192, 198, 200, 204, 216, 228, 236, 238, 240, 242
Offset: 1

Views

Author

Gus Wiseman, May 16 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.
Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

Examples

			The distinct prime indices of 45 are {2,3}, which are the binary indices of 6, which is not prime, so 45 is not in the sequence.
The distinct prime indices of 60 are {1,2,3}, which are the binary indices of 7, which is prime, so 60 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}

Crossrefs

For even instead of prime we have A005408, with multiplicity A003159.
For odd instead of prime we have A005843, with multiplicity A036554.
For prime indices with multiplicity we have A277319, counted by A372688.
The squarefree case is A372851, counted by A372687.
Partitions of this type are counted by A372887.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A096111, A372429, A372436, A372441, A372471, A372689.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeQ[Total[2^(Union[prix[#]]-1)]]&]

Formula

Numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the distinct prime indices of k.
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