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

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

Author

Gus Wiseman, May 04 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.

Note that a number's binary and prime indices cannot have the same minimum; see A372437.

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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[#]]&]

Formula

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