A339841 Numbers that can be factored into distinct primes or semiprimes in exactly one way.

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, 61, 67, 71, 73, 79, 80, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167, 169, 173, 176, 179, 181, 191, 193

Offset: 1

Author

Gus Wiseman, Dec 25 2020

Keywords

• nonn

Comments

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

Examples

			The sequence of terms together with their one factorization begins:
     1 =        29 = 29        80 = 2*4*10
     2 = 2      31 = 31        83 = 83
     3 = 3      37 = 37        89 = 89
     4 = 4      41 = 41        97 = 97
     5 = 5      43 = 43       101 = 101
     7 = 7      47 = 47       103 = 103
     8 = 2*4    48 = 2*4*6    107 = 107
     9 = 9      49 = 49       109 = 109
    11 = 11     53 = 53       112 = 2*4*14
    13 = 13     59 = 59       113 = 113
    17 = 17     61 = 61       121 = 121
    19 = 19     67 = 67       125 = 5*25
    23 = 23     71 = 71       127 = 127
    25 = 25     73 = 73       131 = 131
    27 = 3*9    79 = 79       137 = 137
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Degree Sequence.
• Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

Crossrefs

See link for additional cross-references.

These are the positions of ones in A339839.

The version for no factorizations is A339840.

The version for at least one factorization is A339889.

A001055 counts factorizations.

A001358 lists semiprimes, with squarefree case A006881.

A037143 lists primes and semiprimes.

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

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

A338915 counts partitions that cannot be partitioned into distinct pairs.

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

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]]==1&]