A363123 Primitive terms of A363122: terms k of A363122 such that k/2 is not a term of A363122.

2, 12, 40, 56, 120, 144, 168, 176, 208, 280, 528, 544, 608, 624, 720, 736, 800, 840, 864, 880, 928, 992, 1008, 1040, 1232, 1456, 1584, 1632, 1824, 1872, 2208, 2288, 2368, 2400, 2624, 2640, 2720, 2752, 2784, 2976, 3008, 3040, 3120, 3136, 3392, 3680, 3696, 3776

Offset: 1

Author

Amiram Eldar, May 16 2023

Keywords

• nonn
• easy

Comments

If k is a term of this sequence then k*2^m is a term of A363122 for any m >= 0.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A363122.

Programs

Maple

filter:= proc(n) local F2,Fp,v2,vp, t;
  F2, Fp:= selectremove(t -> t[1]=2, ifactors(n)[2]);
  if Fp = [] then return (n=2) fi;
  v2:= 2^F2[1,2];
  vp:= max(map(t -> t[1]^t[2],Fp));
  v2 > vp and v2/2 <= vp;
end proc:
select(filter, [seq(i,i=2.10000,2)]); # Robert Israel, May 18 2023

Mathematica

q[n_] := Module[{e = IntegerExponent[n, 2]}, 2^e > Max[Power @@@ FactorInteger[n/2^e]]]; Select[Range[2, 10000, 2], q[#] && ! q[#/2] &]

PARI

is1(n) = {my(e = valuation(n, 2), m = n>>e); if(m == 1, n>1, f = factor(m); 2^e > vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); } \\ A363122
is(n) = !(n%2) && is1(n) && !is1(n/2)

Python

from itertools import count, islice
from sympy import factorint
def A363123_gen(startvalue=2): # generator of terms
    return filter(lambda n:(m:=n&-n)>max((p**e for p, e in factorint(n>>(~n&n-1).bit_length()).items()),default=1)>=m>>1,count(max(startvalue,2)))
A363123_list = list(islice(A363123_gen(),20)) # Chai Wah Wu, May 17 2023