A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

1, 3, 5, 15, 27, 39, 55, 63, 85, 121, 125, 135, 169, 171, 175, 209, 243, 247, 255, 299, 375, 399, 437, 459, 507, 539, 605, 637, 725, 735, 783, 841, 867, 891, 1085, 1215, 1323, 1331, 1375, 1519, 1767, 1815, 1863, 2079, 2125, 2187, 2223, 2295, 2299, 2331, 2405

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.

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.

Conjecture: The only number whose binary indices are a subset of its prime indices is 4100, with binary indices {3,13} and prime indices {1,1,3,3,13}. Verified up to 10,000,000.

Examples

			The prime indices of 135 are {2,2,2,3}, and the binary indices are {1,2,3,8}. Since {2,3} is a subset of {1,2,3,8}, 135 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     3: {2}
     5: {3}
    15: {2,3}
    27: {2,2,2}
    39: {2,6}
    55: {3,5}
    63: {2,2,4}
    85: {3,7}
   121: {5,5}
   125: {3,3,3}
The terms together with their binary expansions and binary indices begin:
     1:              1 ~ {1}
     3:             11 ~ {1,2}
     5:            101 ~ {1,3}
    15:           1111 ~ {1,2,3,4}
    27:          11011 ~ {1,2,4,5}
    39:         100111 ~ {1,2,3,6}
    55:         110111 ~ {1,2,3,5,6}
    63:         111111 ~ {1,2,3,4,5,6}
    85:        1010101 ~ {1,3,5,7}
   121:        1111001 ~ {1,4,5,6,7}
   125:        1111101 ~ {1,3,4,5,6,7}
		

Crossrefs

The version for equal lengths is A071814, zeros of A372441.

The version for equal sums is A372427, zeros of A372428.

For disjoint instead of subset we have A372431, complement A372432.

The version for equal maxima is 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, A001221, A230877, A243055, A304818, A355536, A358136, A372429.

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[1000],SubsetQ[bix[#],prix[#]]&]

Formula

Row k of A304038 is a subset of row k of A048793.