cp's OEIS Frontend

A339840 Numbers that cannot be factored into distinct primes or semiprimes.

%I A339840 #24 Feb 12 2021 05:35:23
%S A339840 16,32,64,81,96,128,160,192,224,243,256,288,320,352,384,416,448,486,
%T A339840 512,544,576,608,625,640,704,729,736,768,800,832,864,896,928,960,972,
%U A339840 992,1024,1088,1152,1184,1215,1216,1280,1312,1344,1376,1408,1458,1472,1504
%N A339840 Numbers that cannot be factored into distinct primes or semiprimes.
%C A339840 A semiprime (A001358) is a product of any two prime numbers.
%H A339840 Robert Israel, <a href="/A339840/b339840.txt">Table of n, a(n) for n = 1..10000</a>
%e A339840 The sequence of terms together with their prime indices begins:
%e A339840     16: {1,1,1,1}
%e A339840     32: {1,1,1,1,1}
%e A339840     64: {1,1,1,1,1,1}
%e A339840     81: {2,2,2,2}
%e A339840     96: {1,1,1,1,1,2}
%e A339840    128: {1,1,1,1,1,1,1}
%e A339840    160: {1,1,1,1,1,3}
%e A339840    192: {1,1,1,1,1,1,2}
%e A339840    224: {1,1,1,1,1,4}
%e A339840    243: {2,2,2,2,2}
%e A339840    256: {1,1,1,1,1,1,1,1}
%e A339840    288: {1,1,1,1,1,2,2}
%e A339840    320: {1,1,1,1,1,1,3}
%e A339840    352: {1,1,1,1,1,5}
%e A339840    384: {1,1,1,1,1,1,1,2}
%e A339840    416: {1,1,1,1,1,6}
%e A339840    448: {1,1,1,1,1,1,4}
%e A339840    486: {1,2,2,2,2,2}
%e A339840 For example, a complete list of all factorizations of 192 into primes or semiprimes is:
%e A339840   (2*2*2*2*2*2*3)
%e A339840   (2*2*2*2*2*6)
%e A339840   (2*2*2*2*3*4)
%e A339840   (2*2*2*4*6)
%e A339840   (2*2*3*4*4)
%e A339840   (2*4*4*6)
%e A339840   (3*4*4*4)
%e A339840 Since none of these is strict, 192 is in the sequence.
%p A339840 filter:= proc(n)
%p A339840   g(map(t -> t[2], ifactors(n)[2]))
%p A339840 end proc;
%p A339840 g:= proc(L) option remember; local x,i,j,t,s,Cons,R;
%p A339840   if nops(L) = 1 then return L[1] > 3
%p A339840   elif nops(L) = 2 then return max(L) > 4
%p A339840   fi;
%p A339840   Cons:= {seq(x[i] + x[i,i] + add(x[j,i], j=1..i-1)
%p A339840      + add(x[i,j],j=i+1..nops(L)) = L[i], i=1..nops(L))};
%p A339840   R:= traperror(Optimization:-LPSolve(0,Cons, assume=binary));
%p A339840   type(R,string)
%p A339840 end proc:
%p A339840 select(filter, [$2..2000]); # _Robert Israel_, Dec 28 2020
%t A339840 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A339840 Select[Range[1000],Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]=={}&]
%Y A339840 Allowing only primes gives A013929.
%Y A339840 Removing all squares of primes gives A339740.
%Y A339840 These are the positions of zeros in A339839.
%Y A339840 The complement is A339889.
%Y A339840 A001358 lists semiprimes, with squarefree case A006881.
%Y A339840 A002100 counts partitions into squarefree semiprimes.
%Y A339840 A293511 are a product of distinct squarefree numbers in exactly one way.
%Y A339840 A320663 counts non-isomorphic multiset partitions into singletons or pairs.
%Y A339840 A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339840 A339841 have exactly one factorization into primes or semiprimes.
%Y A339840 The following count factorizations:
%Y A339840 - A001055 into all positive integers > 1.
%Y A339840 - A320655 into semiprimes.
%Y A339840 - A320656 into squarefree semiprimes.
%Y A339840 - A320732 into primes or semiprimes.
%Y A339840 - A322353 into distinct semiprimes.
%Y A339840 - A339661 into distinct squarefree semiprimes.
%Y A339840 - A339742 into distinct primes or squarefree semiprimes.
%Y A339840 - A339839 into distinct primes or semiprimes.
%Y A339840 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339840 - A321728 is conjectured to count non-half-loop-graphical partitions of n.
%Y A339840 - A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A339840 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339840 Cf. A000070, A028260, A320893, A320922, A339741, A339846.
%K A339840 nonn
%O A339840 1,1
%A A339840 _Gus Wiseman_, Dec 20 2020