cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 20 2020

Keywords

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

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/@#]&]=={}&]

A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

Original entry on oeis.org

36, 100, 196, 216, 225, 360, 441, 484, 504, 540, 600, 676, 756, 792, 936, 1000, 1089, 1156, 1176, 1188, 1224, 1225, 1296, 1350, 1368, 1400, 1404, 1444, 1500, 1521, 1656, 1836, 1960, 2052, 2088, 2116, 2160, 2200, 2232, 2250, 2484, 2600, 2601, 2646, 2664, 2744
Offset: 1

Views

Author

Gus Wiseman, Dec 30 2020

Keywords

Comments

Of course, every number is a product of squarefree numbers (A050320).
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
All terms have even Omega (A001222, A028260).

Examples

			The sequence of terms together with their prime indices begins:
      36: {1,1,2,2}        1000: {1,1,1,3,3,3}
     100: {1,1,3,3}        1089: {2,2,5,5}
     196: {1,1,4,4}        1156: {1,1,7,7}
     216: {1,1,1,2,2,2}    1176: {1,1,1,2,4,4}
     225: {2,2,3,3}        1188: {1,1,2,2,2,5}
     360: {1,1,1,2,2,3}    1224: {1,1,1,2,2,7}
     441: {2,2,4,4}        1225: {3,3,4,4}
     484: {1,1,5,5}        1296: {1,1,1,1,2,2,2,2}
     504: {1,1,1,2,2,4}    1350: {1,2,2,2,3,3}
     540: {1,1,2,2,2,3}    1368: {1,1,1,2,2,8}
     600: {1,1,1,2,3,3}    1400: {1,1,1,3,3,4}
     676: {1,1,6,6}        1404: {1,1,2,2,2,6}
     756: {1,1,2,2,2,4}    1444: {1,1,8,8}
     792: {1,1,1,2,2,5}    1500: {1,1,2,3,3,3}
     936: {1,1,1,2,2,6}    1521: {2,2,6,6}
For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:
  7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),
but since none of these is strict, 7560 is in the sequence.

Links

Crossrefs

See link for additional cross references.
The distinct prime shadows (under A181819) of these terms are A339842.
Factorizations into squarefree semiprimes are counted by A320656.
Products of squarefree semiprimes that are not products of distinct semiprimes are A320893.
Factorizations into distinct squarefree semiprimes are A339661.
For the next four lines, we list numbers with even Omega (A028260).
- A320891 cannot be factored into squarefree semiprimes.
- A320894 cannot be factored into distinct squarefree semiprimes.
- A320911 can be factored into squarefree semiprimes.
- A339561 can be factored into distinct squarefree semiprimes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A030229 lists squarefree numbers with even Omega.
A050320 counts factorizations into squarefree numbers.
A050326 counts factorizations into distinct squarefree numbers.
A181819 is the Heinz number of the prime signature of n (prime shadow).
A320656 counts factorizations into squarefree semiprimes.
A339560 can be partitioned into distinct strict pairs.
Cf. A001055, A001222, A005117, A007717, A096373, A112798, A300061, A322353, A339559, A338899, A339740.

Programs

  • Mathematica
    strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]

Formula

Equals A320894 /\ A320911.
Numbers n such that A320656(n) > 0 but A339661(n) = 0.
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