cp's OEIS Frontend

A372851 Squarefree numbers whose prime indices are the binary indices of some prime number.

%I A372851 #7 May 17 2024 19:47:15
%S A372851 3,6,10,22,30,42,46,66,70,102,114,118,130,182,238,246,266,318,330,354,
%T A372851 370,402,406,434,442,510,546,646,654,690,762,770,798,930,938,946,962,
%U A372851 986,1066,1102,1122,1178,1218,1222,1246,1258,1334,1378,1430,1482,1578
%N A372851 Squarefree numbers whose prime indices are the binary indices of some prime number.
%C A372851 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 A372851 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 A372851 Note the function taking a set s to its rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).
%F A372851 Squarefree 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 A372851 The prime indices of 70 are {1,3,4}, which are the binary indices of 13, which is prime, so 70 is in the sequence.
%e A372851 The prime indices of 15 are {2,3}, which are the binary indices of 6, which is not prime, so 15 is not in the sequence.
%e A372851 The terms together with their prime indices begin:
%e A372851     3: {2}
%e A372851     6: {1,2}
%e A372851    10: {1,3}
%e A372851    22: {1,5}
%e A372851    30: {1,2,3}
%e A372851    42: {1,2,4}
%e A372851    46: {1,9}
%e A372851    66: {1,2,5}
%e A372851    70: {1,3,4}
%e A372851   102: {1,2,7}
%e A372851   114: {1,2,8}
%e A372851   118: {1,17}
%e A372851   130: {1,3,6}
%e A372851   182: {1,4,6}
%e A372851   238: {1,4,7}
%e A372851   246: {1,2,13}
%e A372851   266: {1,4,8}
%e A372851   318: {1,2,16}
%e A372851   330: {1,2,3,5}
%e A372851   354: {1,2,17}
%e A372851   370: {1,3,12}
%e A372851   402: {1,2,19}
%t A372851 Select[Range[100],SquareFreeQ[#] && PrimeQ[Total[2^(PrimePi/@First/@FactorInteger[#]-1)]]&]
%Y A372851 [Warning: do not confuse A372887 with the strict case A372687.]
%Y A372851 For odd instead of prime we have A039956.
%Y A372851 For even instead of prime we have A056911.
%Y A372851 Strict partitions of this type are counted by A372687.
%Y A372851 Non-strict partitions of this type are counted by A372688, ranks A277319.
%Y A372851 The nonsquarefree version is A372850, counted by A372887.
%Y A372851 A014499 lists binary indices of prime numbers.
%Y A372851 A019565 gives Heinz number of binary indices, adjoint A048675.
%Y A372851 A038499 counts partitions of prime length, strict A085756.
%Y A372851 A048793 and A272020 (reverse) list binary indices:
%Y A372851 - length A000120
%Y A372851 - min A001511
%Y A372851 - sum A029931
%Y A372851 - max A070939
%Y A372851 A058698 counts partitions of prime numbers, strict A064688.
%Y A372851 A372885 lists primes whose binary indices sum to a prime, indices A372886.
%Y A372851 Cf. A000040, A005940, A025147, A035100, A071814, A096111, A096765, A231204, A372429, A372471, A372689.
%K A372851 nonn
%O A372851 1,1
%A A372851 _Gus Wiseman_, May 16 2024