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

16, 32, 64, 81, 96, 128, 160, 192, 224, 243, 256, 288, 320, 352, 384, 416, 448, 486, 512, 544, 576, 608, 625, 640, 704, 729, 736, 768, 800, 832, 864, 896, 928, 960, 972, 992, 1024, 1088, 1152, 1184, 1215, 1216, 1280, 1312, 1344, 1376, 1408, 1458, 1472, 1504

Offset: 1

Author

Gus Wiseman, Dec 20 2020

Keywords

• nonn

Comments

A semiprime (A001358) is a product of any two prime numbers.

Examples

			The sequence of terms together with their prime indices begins:
    16: {1,1,1,1}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    96: {1,1,1,1,1,2}
   128: {1,1,1,1,1,1,1}
   160: {1,1,1,1,1,3}
   192: {1,1,1,1,1,1,2}
   224: {1,1,1,1,1,4}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   288: {1,1,1,1,1,2,2}
   320: {1,1,1,1,1,1,3}
   352: {1,1,1,1,1,5}
   384: {1,1,1,1,1,1,1,2}
   416: {1,1,1,1,1,6}
   448: {1,1,1,1,1,1,4}
   486: {1,2,2,2,2,2}
For example, a complete list of all factorizations of 192 into primes or semiprimes is:
  (2*2*2*2*2*2*3)
  (2*2*2*2*2*6)
  (2*2*2*2*3*4)
  (2*2*2*4*6)
  (2*2*3*4*4)
  (2*4*4*6)
  (3*4*4*4)
Since none of these is strict, 192 is in the sequence.
		

Links

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

Crossrefs

Allowing only primes gives A013929.

Removing all squares of primes gives A339740.

These are the positions of zeros in A339839.

The complement is A339889.

A001358 lists semiprimes, with squarefree case A006881.

A002100 counts partitions into squarefree semiprimes.

A293511 are a product of distinct squarefree numbers in exactly one way.

A320663 counts non-isomorphic multiset partitions into singletons or pairs.

A338915 cannot be partitioned into distinct pairs (A320892).

A339841 have exactly one factorization into primes or semiprimes.

The following count factorizations:

- A001055 into all positive integers > 1.

- A320655 into semiprimes.

- A320656 into squarefree semiprimes.

- A320732 into primes or semiprimes.

- A322353 into distinct semiprimes.

- A339661 into distinct squarefree semiprimes.

- A339742 into distinct primes or squarefree semiprimes.

- A339839 into distinct primes or semiprimes.

The following count vertex-degree partitions and give their Heinz numbers:

- A321728 is conjectured to count non-half-loop-graphical partitions of n.

- A339617 counts non-graphical partitions of 2n, ranked by A339618.

- A339655 counts non-loop-graphical partitions of 2n (A339657).

Cf. A000070, A028260, A320893, A320922, A339741, A339846.

Programs

Maple

filter:= proc(n)
  g(map(t -> t[2], ifactors(n)[2]))
end proc;
g:= proc(L) option remember; local x,i,j,t,s,Cons,R;
  if nops(L) = 1 then return L[1] > 3
  elif nops(L) = 2 then return max(L) > 4
  fi;
  Cons:= {seq(x[i] + x[i,i] + add(x[j,i], j=1..i-1)
     + add(x[i,j],j=i+1..nops(L)) = L[i], i=1..nops(L))};
  R:= traperror(Optimization:-LPSolve(0,Cons, assume=binary));
  type(R,string)
end proc:
select(filter, [$2..2000]); # Robert Israel, Dec 28 2020

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]=={}&]