cp's OEIS Frontend

A320674 Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [m == 0 (mod prime(i))] for i = 1..k (where prime(i) denotes the i-th prime number and [] is an Iverson bracket).

%I A320674 #16 Nov 09 2018 22:19:48
%S A320674 2,4,6,8,10,12,16,20,24,32,40,48,64,80,96,128,160,192,256,320,384,512,
%T A320674 640,768,1024,1280,1536,2048,2560,3072,4096,5120,6144,8192,10240,
%U A320674 12288,16384,20480,24576,32768,40960,49152,65536,81920,98304,131072,163840,196608
%N A320674 Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [m == 0 (mod prime(i))] for i = 1..k (where prime(i) denotes the i-th prime number and [] is an Iverson bracket).
%C A320674 In other words, the 1's in the binary representation of a term of this sequence encode the first prime divisors of this term.
%C A320674 All terms are even.
%C A320674 All even terms in A029747 belong to this sequence.
%C A320674 The term a(71) = 33554434 is the first one that does not belong to A029747.
%C A320674 See A320673 for similar sequences.
%e A320674 The initial terms, alongside their binary representation and the prime divisors encoded therein, are:
%e A320674   n   a(n)      bin(a(n))                   First prime divisors
%e A320674   --  --------  --------------------------  --------------------
%e A320674    1         2  10                          2
%e A320674    2         4  100                         2
%e A320674    3         6  110                         2, 3
%e A320674    4         8  1000                        2
%e A320674    5        10  1010                        2, 5
%e A320674    6        12  1100                        2, 3
%e A320674    7        16  10000                       2
%e A320674    8        20  10100                       2, 5
%e A320674    9        24  11000                       2, 3
%e A320674   ...
%e A320674   71  33554434  10000000000000000000000010  2, 97
%e A320674   ...
%e A320674 33554434 is in the sequence because its binary expansion 10000000000000000000000010 of length 26 has a 1 in the 1st place and in the 25th place from the left and 0 elsewhere. As it is divisible by the 1st and 25th prime and by no other prime with index <= 26, 33554434 in the sequence. - _David A. Corneth_, Oct 20 2018
%t A320674 selQ[n_] := With[{bb = IntegerDigits[n, 2]}, (Prime /@ Flatten[Position[bb, 1]]) == FactorInteger[n][[All, 1]]];
%t A320674 Select[Range[2, 200000], selQ] (* _Jean-François Alcover_, Nov 01 2018 *)
%o A320674 (PARI) is(n) = my (b=binary(n)); b==vector(#b, k, n%prime(k)==0)
%Y A320674 Cf. A029747, A070939, A320673.
%K A320674 nonn,base
%O A320674 1,1
%A A320674 _Rémy Sigrist_, Oct 19 2018