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).

2, 4, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608

Offset

1,1

Comments

In other words, the 1's in the binary representation of a term of this sequence encode the first prime divisors of this term.

All terms are even.

All even terms in A029747 belong to this sequence.

The term a(71) = 33554434 is the first one that does not belong to A029747.

See A320673 for similar sequences.

Links

Table of n, a(n) for n=1..48.

Example

The initial terms, alongside their binary representation and the prime divisors encoded therein, are:
  n   a(n)      bin(a(n))                   First prime divisors
  --  --------  --------------------------  --------------------
   1         2  10                          2
   2         4  100                         2
   3         6  110                         2, 3
   4         8  1000                        2
   5        10  1010                        2, 5
   6        12  1100                        2, 3
   7        16  10000                       2
   8        20  10100                       2, 5
   9        24  11000                       2, 3
  ...
  71  33554434  10000000000000000000000010  2, 97
  ...
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

Mathematica

selQ[n_] := With[{bb = IntegerDigits[n, 2]}, (Prime /@ Flatten[Position[bb, 1]]) == FactorInteger[n][[All, 1]]];
Select[Range[2, 200000], selQ] (* Jean-François Alcover, Nov 01 2018 *)

Prog

(PARI) is(n) = my (b=binary(n)); b==vector(#b, k, n%prime(k)==0)

Crossrefs

Cf. A029747, A070939, A320673.

Sequence in context: A274284 A375473 A112421 * A022483 A100180 A258137

Adjacent sequences: A320671 A320672 A320673 * A320675 A320676 A320677

Keyword

nonn,base

Author

Rémy Sigrist, Oct 19 2018

Status

approved