cp's OEIS Frontend

A339740 Non-products of distinct primes or squarefree semiprimes.

%I A339740 #12 Jan 05 2021 21:35:10
%S A339740 4,8,9,16,24,25,27,32,40,48,49,54,56,64,72,80,81,88,96,104,108,112,
%T A339740 121,125,128,135,136,144,152,160,162,169,176,184,189,192,200,208,216,
%U A339740 224,232,240,243,248,250,256,272,288,289,296,297,304,320,324,328,336
%N A339740 Non-products of distinct primes or squarefree semiprimes.
%C A339740 Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.
%e A339740 The sequence of terms together with their prime indices begins:
%e A339740       4: {1,1}             80: {1,1,1,1,3}
%e A339740       8: {1,1,1}           81: {2,2,2,2}
%e A339740       9: {2,2}             88: {1,1,1,5}
%e A339740      16: {1,1,1,1}         96: {1,1,1,1,1,2}
%e A339740      24: {1,1,1,2}        104: {1,1,1,6}
%e A339740      25: {3,3}            108: {1,1,2,2,2}
%e A339740      27: {2,2,2}          112: {1,1,1,1,4}
%e A339740      32: {1,1,1,1,1}      121: {5,5}
%e A339740      40: {1,1,1,3}        125: {3,3,3}
%e A339740      48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
%e A339740      49: {4,4}            135: {2,2,2,3}
%e A339740      54: {1,2,2,2}        136: {1,1,1,7}
%e A339740      56: {1,1,1,4}        144: {1,1,1,1,2,2}
%e A339740      64: {1,1,1,1,1,1}    152: {1,1,1,8}
%e A339740      72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
%e A339740 For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.
%t A339740 sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t A339740 Select[Range[100],sqps[#]=={}&]
%Y A339740 A013929 allows only primes.
%Y A339740 A320894 does not allow primes (but omega is assumed even).
%Y A339740 A339741 is the complement.
%Y A339740 A339742 has zeros at these positions.
%Y A339740 A339840 allows squares of primes.
%Y A339740 A001358 lists semiprimes, with squarefree case A006881.
%Y A339740 A002100 counts partitions into squarefree semiprimes.
%Y A339740 A320663 counts non-isomorphic multiset partitions into singletons or pairs.
%Y A339740 A339841 have exactly one factorization into primes or semiprimes.
%Y A339740 The following count factorizations:
%Y A339740 - A001055 into all positive integers > 1.
%Y A339740 - A050326 into distinct squarefree numbers.
%Y A339740 - A320655 into semiprimes.
%Y A339740 - A320656 into squarefree semiprimes.
%Y A339740 - A320732 into primes or semiprimes.
%Y A339740 - A322353 into distinct semiprimes.
%Y A339740 - A339661 into distinct squarefree semiprimes.
%Y A339740 - A339839 into distinct primes or semiprimes.
%Y A339740 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339740 - A058696 counts partitions of 2n (A300061).
%Y A339740 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339740 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339740 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339740 - A321728 is conjectured to count non-half-loop-graphical partitions of n.
%Y A339740 The following count partitions/factorizations of even length and give their Heinz numbers:
%Y A339740 - A027187/A339846 counts all of even length (A028260).
%Y A339740 - A096373/A339737 cannot be partitioned into strict pairs (A320891).
%Y A339740 - A338915/A339662 cannot be partitioned into distinct pairs (A320892).
%Y A339740 - A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).
%Y A339740 Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.
%K A339740 nonn
%O A339740 1,1
%A A339740 _Gus Wiseman_, Dec 20 2020