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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.
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%I #18 Oct 26 2020 08:06:12

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

%N 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.

%C 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.

%H Jan-Christoph Schlage-Puchta, <a href="http://dx.doi.org/10.1007/s00013-016-0963-6">The exponents in the prime decomposition of factorials</a>, Archiv der Mathematik 107:6 (2016), pp. 603-608.

%H Vladimir Shevelev, <a href="https://doi.org/10.4064/aa126-3-1">Compact integers and factorials</a>, Acta Arithmetica 126 (2007), pp. 195-236.

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

%p filter:= proc(n)

%p local a;

%p a:= padic:-ordp(n!,2);

%p if a > 0 and a <> 2^padic:-ordp(a,2) then return false fi;

%p a:= padic:-ordp(n!,3);

%p a = 0 or a = 2^padic:-ordp(a,2)

%p end proc:

%p select(filter, [$1..20]); # _Robert Israel_, Dec 05 2016

%t 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]];

%t Select[Range[20], filterQ] (* _Jean-François Alcover_, Oct 26 2020, after Maple *)

%K nonn,fini,full

%O 1,2

%A _Charles R Greathouse IV_, Dec 02 2016