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

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

Author

Gus Wiseman, May 14 2024

Keywords

• nonn

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.