cp's OEIS Frontend

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

%I A371290 #9 Mar 28 2024 11:58:06
%S A371290 1,2,3,4,5,8,9,10,11,16,17,64,65,128,129,130,131,136,137,138,139,256,
%T A371290 257,260,261,1024,1025,4096,4097,32768,32769,32770,32771,32776,32777,
%U A371290 32778,32779,32896,32897,32898,32899,32904,32905,32906,32907,65536,65537,262144
%N A371290 Numbers whose product of binary indices is a prime power > 1.
%C A371290 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 A371290 The terms together with their binary expansions and binary indices begin:
%e A371290        1:                   1 ~ {1}
%e A371290        2:                  10 ~ {2}
%e A371290        3:                  11 ~ {1,2}
%e A371290        4:                 100 ~ {3}
%e A371290        5:                 101 ~ {1,3}
%e A371290        8:                1000 ~ {4}
%e A371290        9:                1001 ~ {1,4}
%e A371290       10:                1010 ~ {2,4}
%e A371290       11:                1011 ~ {1,2,4}
%e A371290       16:               10000 ~ {5}
%e A371290       17:               10001 ~ {1,5}
%e A371290       64:             1000000 ~ {7}
%e A371290       65:             1000001 ~ {1,7}
%e A371290      128:            10000000 ~ {8}
%e A371290      129:            10000001 ~ {1,8}
%e A371290      130:            10000010 ~ {2,8}
%e A371290      131:            10000011 ~ {1,2,8}
%e A371290      136:            10001000 ~ {4,8}
%e A371290      137:            10001001 ~ {1,4,8}
%e A371290      138:            10001010 ~ {2,4,8}
%e A371290      139:            10001011 ~ {1,2,4,8}
%e A371290      256:           100000000 ~ {9}
%e A371290      257:           100000001 ~ {1,9}
%e A371290      260:           100000100 ~ {3,9}
%e A371290      261:           100000101 ~ {1,3,9}
%e A371290     1024:         10000000000 ~ {11}
%e A371290     1025:         10000000001 ~ {1,11}
%e A371290     4096:       1000000000000 ~ {13}
%e A371290     4097:       1000000000001 ~ {1,13}
%e A371290    32768:    1000000000000000 ~ {16}
%t A371290 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371290 Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]
%Y A371290 For powers of 2 we have A253317.
%Y A371290 For prime indices we have A320698.
%Y A371290 For squarefree numbers instead of prime powers we have A371289.
%Y A371290 A000040 lists prime numbers.
%Y A371290 A000961 lists prime-powers.
%Y A371290 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A371290 A070939 gives length of binary expansion.
%Y A371290 A096111 gives product of binary indices.
%Y A371290 Cf. A005117, A326782, A368533, A371292, A371443, A371452.
%K A371290 nonn,base
%O A371290 1,2
%A A371290 _Gus Wiseman_, Mar 27 2024