cp's OEIS Frontend

A372850 Numbers whose distinct prime indices are the binary indices of some prime number.

%I A372850 #6 May 17 2024 19:47:20
%S A372850 3,6,9,10,12,18,20,22,24,27,30,36,40,42,44,46,48,50,54,60,66,70,72,80,
%T A372850 81,84,88,90,92,96,100,102,108,114,118,120,126,130,132,140,144,150,
%U A372850 160,162,168,176,180,182,184,192,198,200,204,216,228,236,238,240,242
%N A372850 Numbers whose distinct prime indices are the binary indices of some prime number.
%C A372850 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.
%C A372850 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.
%C A372850 Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).
%F A372850 Numbers k such that Sum_{i:prime(i)|k} 2^(i-1) is prime, where the sum is over the distinct prime indices of k.
%e A372850 The distinct prime indices of 45 are {2,3}, which are the binary indices of 6, which is not prime, so 45 is not in the sequence.
%e A372850 The distinct prime indices of 60 are {1,2,3}, which are the binary indices of 7, which is prime, so 60 is in the sequence.
%e A372850 The terms together with their prime indices begin:
%e A372850     3: {2}
%e A372850     6: {1,2}
%e A372850     9: {2,2}
%e A372850    10: {1,3}
%e A372850    12: {1,1,2}
%e A372850    18: {1,2,2}
%e A372850    20: {1,1,3}
%e A372850    22: {1,5}
%e A372850    24: {1,1,1,2}
%e A372850    27: {2,2,2}
%e A372850    30: {1,2,3}
%e A372850    36: {1,1,2,2}
%e A372850    40: {1,1,1,3}
%e A372850    42: {1,2,4}
%e A372850    44: {1,1,5}
%e A372850    46: {1,9}
%e A372850    48: {1,1,1,1,2}
%e A372850    50: {1,3,3}
%e A372850    54: {1,2,2,2}
%e A372850    60: {1,1,2,3}
%e A372850    66: {1,2,5}
%e A372850    70: {1,3,4}
%t A372850 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A372850 Select[Range[100],PrimeQ[Total[2^(Union[prix[#]]-1)]]&]
%Y A372850 For even instead of prime we have A005408, with multiplicity A003159.
%Y A372850 For odd instead of prime we have  A005843, with multiplicity A036554.
%Y A372850 For prime indices with multiplicity we have A277319, counted by A372688.
%Y A372850 The squarefree case is A372851, counted by A372687.
%Y A372850 Partitions of this type are counted by A372887.
%Y A372850 A014499 lists binary indices of prime numbers.
%Y A372850 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372850 A038499 counts partitions of prime length, strict A085756.
%Y A372850 A048793 and A272020 (reverse) list binary indices:
%Y A372850 - length A000120
%Y A372850 - min A001511
%Y A372850 - sum A029931
%Y A372850 - max A070939
%Y A372850 A058698 counts partitions of prime numbers, strict A064688.
%Y A372850 A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y A372850 Cf. A000040, A005940, A023506, A029837, A035100, A071814, A096111, A372429, A372436, A372441, A372471, A372689.
%K A372850 nonn
%O A372850 1,1
%A A372850 _Gus Wiseman_, May 16 2024