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

Author

Gus Wiseman, May 02 2024

Keywords

• nonn
• base

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

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

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)).