A371290 Numbers whose product of binary indices is a prime power > 1.

1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 64, 65, 128, 129, 130, 131, 136, 137, 138, 139, 256, 257, 260, 261, 1024, 1025, 4096, 4097, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 65536, 65537, 262144

Offset: 1

Author

Gus Wiseman, Mar 27 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.

Examples

			The terms together with their binary expansions and binary indices begin:
       1:                   1 ~ {1}
       2:                  10 ~ {2}
       3:                  11 ~ {1,2}
       4:                 100 ~ {3}
       5:                 101 ~ {1,3}
       8:                1000 ~ {4}
       9:                1001 ~ {1,4}
      10:                1010 ~ {2,4}
      11:                1011 ~ {1,2,4}
      16:               10000 ~ {5}
      17:               10001 ~ {1,5}
      64:             1000000 ~ {7}
      65:             1000001 ~ {1,7}
     128:            10000000 ~ {8}
     129:            10000001 ~ {1,8}
     130:            10000010 ~ {2,8}
     131:            10000011 ~ {1,2,8}
     136:            10001000 ~ {4,8}
     137:            10001001 ~ {1,4,8}
     138:            10001010 ~ {2,4,8}
     139:            10001011 ~ {1,2,4,8}
     256:           100000000 ~ {9}
     257:           100000001 ~ {1,9}
     260:           100000100 ~ {3,9}
     261:           100000101 ~ {1,3,9}
    1024:         10000000000 ~ {11}
    1025:         10000000001 ~ {1,11}
    4096:       1000000000000 ~ {13}
    4097:       1000000000001 ~ {1,13}
   32768:    1000000000000000 ~ {16}
		

Crossrefs

For powers of 2 we have A253317.

For prime indices we have A320698.

For squarefree numbers instead of prime powers we have A371289.

A000040 lists prime numbers.

A000961 lists prime-powers.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A005117, A326782, A368533, A371292, A371443, A371452.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]