A339841 - OEIS

%I #13 Feb 07 2021 06:25:28

%S 1,2,3,4,5,7,8,9,11,13,17,19,23,25,27,29,31,37,41,43,47,48,49,53,59,

%T 61,67,71,73,79,80,83,89,97,101,103,107,109,112,113,121,125,127,131,

%U 137,139,144,149,151,157,162,163,167,169,173,176,179,181,191,193

%N Numbers that can be factored into distinct primes or semiprimes in exactly one way.

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

%H Amiram Eldar, <a href="/A339841/b339841.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>

%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>

%e The sequence of terms together with their one factorization begins:

%e      1 =        29 = 29        80 = 2*4*10

%e      2 = 2      31 = 31        83 = 83

%e      3 = 3      37 = 37        89 = 89

%e      4 = 4      41 = 41        97 = 97

%e      5 = 5      43 = 43       101 = 101

%e      7 = 7      47 = 47       103 = 103

%e      8 = 2*4    48 = 2*4*6    107 = 107

%e      9 = 9      49 = 49       109 = 109

%e     11 = 11     53 = 53       112 = 2*4*14

%e     13 = 13     59 = 59       113 = 113

%e     17 = 17     61 = 61       121 = 121

%e     19 = 19     67 = 67       125 = 5*25

%e     23 = 23     71 = 71       127 = 127

%e     25 = 25     73 = 73       131 = 131

%e     27 = 3*9    79 = 79       137 = 137

%e For example, we have 360 = 2*3*6*10, so 360 is in the sequence. But 360 is absent from A293511, because we also have 360 = 2*6*30.

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Select[Range[100],Length[Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]]==1&]

%Y See link for additional cross-references.

%Y These are the positions of ones in A339839.

%Y The version for no factorizations is A339840.

%Y The version for at least one factorization is A339889.

%Y A001055 counts factorizations.

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A037143 lists primes and semiprimes.

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

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

%Y A338915 counts partitions that cannot be partitioned into distinct pairs.

%Y Cf. A002494, A013929, A028260, A320893, A320922, A339618, A339740, A339846.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 25 2020