A372437 -id:A372437 - cp's OEIS frontend

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

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

Gus Wiseman, May 02 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.
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

			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}

John Tyler Rascoe, Table of n, a(n) for n = 1..9438

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.

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

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

A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

Original entry on oeis.org

1, 0, 1, -1, 1, 0, 2, -2, 0, 0, 2, -1, 2, 1, 2, -3, 1, -1, 2, -1, 1, 1, 3, -2, 1, 1, 1, 0, 3, 1, 4, -4, 0, 0, 1, -2, 2, 1, 2, -2, 2, 0, 3, 0, 1, 2, 4, -3, 1, 0, 2, 0, 3, 0, 3, -1, 2, 2, 4, 0, 4, 3, 3, -5, 0, -1, 2, -1, 1, 0, 3, -3, 2, 1, 1, 0, 2, 1, 4, -3, -1

Offset: 1

Gus Wiseman, May 07 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.

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.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of zeros are A071814.
For sum instead of length we have A372428, zeros A372427.
For minimum instead of length we have A372437, zeros {}.
For maximum instead of length we have A372442, zeros A372436.
Positions of odd terms are A372590, even A372591.
A003963 gives product of prime indices.
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, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.

f:= proc(n) convert(convert(n,base,2),`+`)-numtheory:-bigomega(n) end proc:
map(f, [$1..100]); # Robert Israel, May 22 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[bix[n]]-Length[prix[n]],{n,100}]

a(n) = A000120(n) - A001222(n).

A372436 Numbers whose binary indices and prime indices have the same maximum.

Original entry on oeis.org

3, 5, 14, 22, 39, 52, 68, 85, 102, 119, 133, 152, 171, 190, 209, 228, 247, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 928, 957, 986, 1015, 1054, 1085, 1116, 1178, 1209, 1240, 1302

Offset: 1

Gus Wiseman, May 04 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.
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 that a number's binary and prime indices cannot have the same minimum; see A372437.

			The binary indices of 345 are {1,4,5,7,9}, and the prime indices are {2,3,9}. Both have maximum 9, so 345 is in the sequence.
The terms together with their prime indices begin:
     3: {2}
     5: {3}
    14: {1,4}
    22: {1,5}
    39: {2,6}
    52: {1,1,6}
    68: {1,1,7}
    85: {3,7}
   102: {1,2,7}
   119: {4,7}
   133: {4,8}
   152: {1,1,1,8}
   171: {2,2,8}
The terms together with their binary expansions and binary indices begin:
     3:           11 ~ {1,2}
     5:          101 ~ {1,3}
    14:         1110 ~ {2,3,4}
    22:        10110 ~ {2,3,5}
    39:       100111 ~ {1,2,3,6}
    52:       110100 ~ {3,5,6}
    68:      1000100 ~ {3,7}
    85:      1010101 ~ {1,3,5,7}
   102:      1100110 ~ {2,3,6,7}
   119:      1110111 ~ {1,2,3,5,6,7}
   133:     10000101 ~ {1,3,8}
   152:     10011000 ~ {4,5,8}
   171:     10101011 ~ {1,2,4,6,8}

For length instead of maximum we have A071814.
For sum instead of maximum we have A372427.
Positions of zeros in A372442, for minimum instead of maximum A372437.
A003963 gives product of prime indices.
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, A066099, A096111, A304818, A355536, A359401, A359402, A372428-A372433, A372441.

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

A070939(a(n)) = A061395(a(n)).

A372428 Sum of binary indices of n minus sum of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 4, 5, 1, -1, 2, 0, 3, 3, 4, 2, 4, 4, 4, 6, 6, 3, 8, 4, 1, 0, 0, 2, 3, -2, 2, 4, 4, -2, 5, -1, 6, 7, 5, 1, 5, 4, 6, 5, 6, -1, 9, 9, 8, 6, 6, 1, 11, 1, 8, 13, 1, -1, 1, -9, 1, 0, 4, -7, 4, -9, 0, 6, 4, 6, 7, -5, 5, 5, 0, -8

Offset: 1

Gus Wiseman, May 02 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.

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 binary indices of 65 are {1,7}, and the prime indices are {3,6}, so a(65) = 8 - 9 = -1.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Positions of zeros are A372427.
For minimum instead of sum we have A372437.
For length instead of sum we have A372441, zeros A071814.
For maximum instead of sum we have A372442, zeros A372436.
Positions of odd terms are A372586, even A372587.
A003963 gives product of prime indices.
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.
A096111 gives product of binary indices.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
A326031 gives weight of the set-system with BII-number n.
Cf. A000720, A001221, A014499, A030101, A059893, A066099, A243055, A304818, A359495, A372429-A372432.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[bix[n]]-Total[prix[n]],{n,100}]

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    for n in count(1):
        b = sum((i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1')
        p = sum(sieve.search(i)[0] for i in factorint(n, multiple=True))
        yield(b-p)
A372428_list = list(islice(a_gen(), 83)) # John Tyler Rascoe, May 04 2024

from sympy import primepi, factorint
def A372428(n): return int(sum(i for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')-sum(primepi(p)*e for p, e in factorint(n).items())) # Chai Wah Wu, Oct 18 2024

a(n) = A029931(n) - A056239(n).

A372442 (Greatest binary index of n) minus (greatest prime index of n).

Original entry on oeis.org

1, 0, 2, 0, 1, -1, 3, 2, 1, -1, 2, -2, 0, 1, 4, -2, 3, -3, 2, 1, 0, -4, 3, 2, -1, 3, 1, -5, 2, -6, 5, 1, -1, 2, 4, -6, -2, 0, 3, -7, 2, -8, 1, 3, -3, -9, 4, 2, 3, -1, 0, -10, 4, 1, 2, -2, -4, -11, 3, -12, -5, 2, 6, 1, 2, -12, 0, -2, 3, -13, 5, -14, -5, 4, -1

Offset: 2

Gus Wiseman, May 07 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.

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.

For sum instead of maximum we have A372428, zeros A372427.
Positions of zeros are A372436.
For minimum instead of maximum we have A372437, zeros {}.
For length instead of maximum we have A372441, zeros A071814.
Positions of odd terms are A372588, even A372589.
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, A003963, A014499, A174090, A243055, A355536, A359495, A372429-A372432, A372589-A372591.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max[bix[n]]-Max[prix[n]],{n,2,100}]

a(n) = A070939(n) - A061395(n) = A029837(n) - A061395(n) for n > 1.

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

Original entry on oeis.org

2, 6, 7, 8, 9, 10, 11, 13, 15, 19, 24, 28, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 54, 57, 58, 59, 60, 61, 65, 67, 70, 73, 76, 77, 79, 85, 86, 90, 95, 96, 97, 98, 103, 106, 107, 109, 110, 111, 112, 117, 119, 123, 124, 126, 127, 128, 129

Offset: 1

Gus Wiseman, May 14 2024

nonn

The odd version is A372590.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
          {2}   2  (1)
        {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)
      {1,2,4}  11  (5)
      {1,3,4}  13  (6)
    {1,2,3,4}  15  (3,2)
      {1,2,5}  19  (8)
        {4,5}  24  (2,1,1,1)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
          {6}  32  (1,1,1,1,1)
        {1,6}  33  (5,2)
        {2,6}  34  (7,1)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
    {1,2,3,6}  39  (6,2)
        {4,6}  40  (3,1,1,1)
      {1,4,6}  41  (13)
      {2,4,6}  42  (4,2,1)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
Positions of even terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372590.
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.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],EvenQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

Original entry on oeis.org

2, 6, 7, 8, 10, 11, 15, 18, 19, 21, 24, 26, 27, 28, 29, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 65, 70, 71, 72, 74, 76, 78, 79, 81, 84, 86, 87, 89, 91, 95, 96, 98, 101, 104, 105, 106, 107, 108, 111, 112, 113, 114, 116, 117, 122, 126, 128

Offset: 1

Gus Wiseman, May 14 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.
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 A372589.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {2}   2  (1)
      {2,3}   6  (2,1)
    {1,2,3}   7  (4)
        {4}   8  (1,1,1)
      {2,4}  10  (3,1)
    {1,2,4}  11  (5)
  {1,2,3,4}  15  (3,2)
      {2,5}  18  (2,2,1)
    {1,2,5}  19  (8)
    {1,3,5}  21  (4,2)
      {4,5}  24  (2,1,1,1)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
    {3,4,5}  28  (4,1,1)
  {1,3,4,5}  29  (10)
        {6}  32  (1,1,1,1,1)
      {1,6}  33  (5,2)
      {2,6}  34  (7,1)
      {4,6}  40  (3,1,1,1)
    {1,4,6}  41  (13)
    {3,4,6}  44  (5,1,1)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
Positions of odd terms in A372442, zeros A372436.
The complement is A372589.
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, A006141, A066208, A160786, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],OddQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

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

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

Gus Wiseman, May 14 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.
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.

			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)

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.

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[#]]]&]

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

Gus Wiseman, May 14 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.
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.

			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)

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.

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

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

Gus Wiseman, May 14 2024

nonn

The even version is A372591.

			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)

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.

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

cp's OEIS Frontend

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

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A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

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A372436 Numbers whose binary indices and prime indices have the same maximum.

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A372428 Sum of binary indices of n minus sum of prime indices of n.

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A372442 (Greatest binary index of n) minus (greatest prime index of n).

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A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

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A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

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

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