A365512 a(n) is the least odd prime p such that A000120(n*p) = A000120(n) * A000120(p).

3, 3, 5, 3, 3, 5, 17, 3, 3, 3, 17, 5, 17, 17, 17, 3, 3, 3, 5, 3, 3, 17, 257, 5, 5, 17, 257, 17, 257, 17, 257, 3, 3, 3, 5, 3, 3, 5, 257, 3, 3, 3, 257, 17, 17, 257, 257, 5, 5, 5, 5, 17, 257, 257, 257, 17, 257, 257, 257, 17, 257, 257, 257, 3, 3, 3, 5, 3, 3, 5, 257, 3, 3, 3, 17, 5, 257, 257, 257, 3

Offset: 1

Author

Robert Israel, Sep 07 2023

Keywords

• nonn
• base

Comments

a(2*n) = a(n).

a(n) <= A365475(A070939(n)), with a(2^n-1) = A365475(n).

Examples

			a(3) = 5 because 5 is an odd prime with A000120(3 * 5) = 4 = A000120(3) * A000120(5) while A000120(3 * 3) = 2 < 4 = A000120(3) * A000120(3).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A070939, A365475.

Programs

Maple

g:= n -> convert(convert(n,base,2),`+`):
f:= proc(n) option remember; local t,S,d,L,B,forbid,i,j,r,q;
  if n::even then return procname(n/2^padic:-ordp(n,2)) fi;
  L:= convert(n,base,2);
  t:= convert(L,`+`);
  B:= select(t -> L[t]=1, [$1..nops(L)]);
  forbid:= {seq(seq(B[i]-B[j],j=1..i-1),i=1..nops(B))};
  S[0]:= [1];
  for d from 1 do
    S[d]:= NULL;
    for j from 0 to d-1 do
      if member(d-j,forbid) then next fi;
      for r in S[j] do
        q:= r + 2^d;
        if g(q*n) = t*g(q) then
          if isprime(q) then return q fi;
          S[d]:= S[d], q;
        fi
      od
    od;
    S[d]:= [S[d]];
  od
end proc:
map(f, [$1..100]);

PARI

a(n) = my(p=3, h=hammingweight(n)); while (hammingweight(n*p) != h*hammingweight(p), p = nextprime(p+1)); p; \\ Michel Marcus, Sep 08 2023