A371290 - OEIS

%I #9 Mar 28 2024 11:58:06

%S 1,2,3,4,5,8,9,10,11,16,17,64,65,128,129,130,131,136,137,138,139,256,

%T 257,260,261,1024,1025,4096,4097,32768,32769,32770,32771,32776,32777,

%U 32778,32779,32896,32897,32898,32899,32904,32905,32906,32907,65536,65537,262144

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

%C 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.

%e The terms together with their binary expansions and binary indices begin:

%e        1:                   1 ~ {1}

%e        2:                  10 ~ {2}

%e        3:                  11 ~ {1,2}

%e        4:                 100 ~ {3}

%e        5:                 101 ~ {1,3}

%e        8:                1000 ~ {4}

%e        9:                1001 ~ {1,4}

%e       10:                1010 ~ {2,4}

%e       11:                1011 ~ {1,2,4}

%e       16:               10000 ~ {5}

%e       17:               10001 ~ {1,5}

%e       64:             1000000 ~ {7}

%e       65:             1000001 ~ {1,7}

%e      128:            10000000 ~ {8}

%e      129:            10000001 ~ {1,8}

%e      130:            10000010 ~ {2,8}

%e      131:            10000011 ~ {1,2,8}

%e      136:            10001000 ~ {4,8}

%e      137:            10001001 ~ {1,4,8}

%e      138:            10001010 ~ {2,4,8}

%e      139:            10001011 ~ {1,2,4,8}

%e      256:           100000000 ~ {9}

%e      257:           100000001 ~ {1,9}

%e      260:           100000100 ~ {3,9}

%e      261:           100000101 ~ {1,3,9}

%e     1024:         10000000000 ~ {11}

%e     1025:         10000000001 ~ {1,11}

%e     4096:       1000000000000 ~ {13}

%e     4097:       1000000000001 ~ {1,13}

%e    32768:    1000000000000000 ~ {16}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]

%Y For powers of 2 we have A253317.

%Y For prime indices we have A320698.

%Y For squarefree numbers instead of prime powers we have A371289.

%Y A000040 lists prime numbers.

%Y A000961 lists prime-powers.

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

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

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

%K nonn,base

%O 1,2

%A _Gus Wiseman_, Mar 27 2024