A214620 Numbers n such that at least one other integer m exists with the same smallest prime factor, same largest prime factor, and same set of binary digits as n.

18, 24, 36, 42, 48, 56, 70, 72, 84, 90, 96, 98, 112, 120, 135, 140, 144, 150, 154, 168, 170, 175, 180, 182, 186, 192, 196, 198, 204, 220, 224, 225, 234, 240, 242, 245, 248, 266, 270, 276, 279, 280, 286, 288, 294, 300, 304, 306, 308, 310, 312, 315, 322, 330, 336, 338

Offset

1,1

Comments

Binary digits of m are a permutation of binary digits of n.

Conjecture: there is X such that among integers bigger than X more than 50% are in the sequence.

Since a set is an unordered collection of distinct elements, one should say "same multiset (or bag) of binary digits as n." - Daniel Forgues, Mar 31 2016

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

18 and 24 have the same set of binary digits: 10010 and 11000, same smallest prime factor 2, and same largest prime factor 3, so both 18 and 24 are in the sequence.

Maple

N:= 10: # to get all terms < 2^N
H:= proc(n) local F, B;
  F:= numtheory:-factorset(n);
B:= sort(convert(n, base, 2));
[min(F), max(F), op(B)];
end proc:
T:= {}:
for n from 1 to 2^N-1 do
h:= H(n);
if assigned(R[h]) then T:= T union {n, R[h]}
else R[h]:= n
fi
od:
sort(convert(T, list)); # Robert Israel, Apr 01 2016

Mathematica

nn = 360; TakeWhile[Union@ Flatten@ Table[Select[Complement[Range[3 n], {n}], And[Length@ Union[DigitCount[#, 2] & /@ {#, n}] == 1, Length@ Union[{First@ #, Last@ #} &@ Map[First, FactorInteger@ #] & /@ {#, n}] == 1] &] /. i_ /; MissingQ@ i -> Nothing, {n, nn}], # <= nn &] (* Michael De Vlieger, Apr 01 2016, Version 10.2 *)

Crossrefs

Cf. A214619, A214621.

Sequence in context: A334015 A162331 A092536 * A360768 A359409 A362432

Adjacent sequences: A214617 A214618 A214619 * A214621 A214622 A214623

Keyword

nonn,base

Author

Alex Ratushnyak, Jul 23 2012

Status

approved