A372427 Numbers whose binary indices and prime indices have the same sum.

19, 33, 34, 69, 74, 82, 130, 133, 305, 412, 428, 436, 533, 721, 755, 808, 917, 978, 1036, 1058, 1062, 1121, 1133, 1143, 1341, 1356, 1630, 1639, 1784, 1807, 1837, 1990, 2057, 2115, 2130, 2133, 2163, 2260, 2324, 2328, 2354, 2358, 2512, 2534, 2627, 2771, 2825

Offset: 1

Author

Gus Wiseman, May 01 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.

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 130 are {2,8}, and the prime indices are {1,3,6}. Both sum to 10, so 130 is in the sequence.
The terms together with their prime indices begin:
   19: {8}
   33: {2,5}
   34: {1,7}
   69: {2,9}
   74: {1,12}
   82: {1,13}
  130: {1,3,6}
  133: {4,8}
  305: {3,18}
  412: {1,1,27}
  428: {1,1,28}
The terms together with their binary expansions and binary indices begin:
   19:      10011 ~ {1,2,5}
   33:     100001 ~ {1,6}
   34:     100010 ~ {2,6}
   69:    1000101 ~ {1,3,7}
   74:    1001010 ~ {2,4,7}
   82:    1010010 ~ {2,5,7}
  130:   10000010 ~ {2,8}
  133:   10000101 ~ {1,3,8}
  305:  100110001 ~ {1,5,6,9}
  412:  110011100 ~ {3,4,5,8,9}
  428:  110101100 ~ {3,4,6,8,9}
		

Links

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

Crossrefs

For length instead of sum we get A071814.

Positions of zeros in A372428.

For maximum instead of sum we have A372436.

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, A066099, A304818, A318283, A355536, A359401, A359402, A372429-A372433, A372441.

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