A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Author

Gus Wiseman, Nov 24 2020

Keywords

• nonn

Comments

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Examples

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6
		

Crossrefs

A002100 counts partitions into squarefree semiprimes.

A056768 uses primes instead of squarefree semiprimes.

A101048 counts partitions into semiprimes.

A338902 is the not necessarily squarefree version.

A339113 includes the Heinz numbers of these partitions.

A001358 lists semiprimes, with odd and even terms A046315 and A100484.

A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.

A320656 counts factorizations into squarefree semiprimes.

A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.

A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.

Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

Programs

Mathematica

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

Formula

a(n) = A002100(A006881(n)).