A278965 Numbers k such that k! = 2^a * 3^b * c, where a and b are 0 or powers of 2 and c is relatively prime to 6.

1, 2, 3, 6, 7, 10, 11, 18, 19

Offset

1,2

Comments

Shevelev proves that this sequence contains no other members. Jan-Christoph Schlage-Puchta proves that "a and b are 0 or powers of 2" can be generalized to "a is 0 or a power of 2 and b is 0 or 3-smooth" without changing the sequence.

Links

Table of n, a(n) for n=1..9.

Jan-Christoph Schlage-Puchta, The exponents in the prime decomposition of factorials, Archiv der Mathematik 107:6 (2016), pp. 603-608.

Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126 (2007), pp. 195-236.

Example

11! = 2^8 * 3^4 * 5^2 * 7 * 11 and 8 and 4 are powers of 2, so 11 is in this sequence.

Maple

filter:= proc(n)
  local a;
  a:= padic:-ordp(n!, 2);
  if a > 0 and a <> 2^padic:-ordp(a, 2) then return false fi;
  a:= padic:-ordp(n!, 3);
  a = 0 or a = 2^padic:-ordp(a, 2)
end proc:
select(filter, [$1..20]); # Robert Israel, Dec 05 2016

Mathematica

filterQ[n_] := Module[{a}, a = IntegerExponent[n!, 2]; If[a > 0 && a != 2^IntegerExponent[a, 2], Return[False]]; a = IntegerExponent[n!, 3]; a == 0 || a == 2^IntegerExponent[a, 2]];
Select[Range[20], filterQ] (* Jean-François Alcover, Oct 26 2020, after Maple *)

Crossrefs

Sequence in context: A345251 A349257 A204323 * A032858 A181498 A030703

Adjacent sequences: A278962 A278963 A278964 * A278966 A278967 A278968

Keyword

nonn,fini,full

Author

Charles R Greathouse IV, Dec 02 2016

Status

approved