A379305 - OEIS

%I #5 Dec 27 2024 18:08:11

%S 0,0,1,2,1,1,2,3,3,3,3,6,8,8,8,10,12,17,18,18,22,28,30,36,40,44,52,62,

%T 67,78,87,97,113,129,137,156,177,200,227,251,271,312,350,382,425,475,

%U 521,588,648,705,785,876,957,1061,1164,1272,1411,1558,1693,1866

%N Number of strict integer partitions of n with a unique prime part.

%e The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):

%e   (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)

%e        (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)

%e                               (421)  (431)  (621)  (631)  (74)   (B1)

%e                                                           (83)   (642)

%e                                                           (92)   (651)

%e                                                           (821)  (741)

%e                                                                  (831)

%e                                                                  (921)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?PrimeQ]==1&]],{n,0,30}]

%Y For all prime parts we have A000586, non-strict A000607 (ranks A076610).

%Y For no prime parts we have A096258, non-strict A002095 (ranks A320628).

%Y Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.

%Y For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).

%Y The non-strict version is A379304.

%Y For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).

%Y Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).

%Y A000040 lists the prime numbers, differences A001223.

%Y A000041 counts integer partitions, strict A000009.

%Y A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.

%Y A095195 gives k-th differences of prime numbers.

%Y Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 27 2024