A379304 - OEIS

%I #5 Dec 27 2024 18:08:15

%S 0,0,1,2,2,3,4,6,7,9,11,17,20,26,31,41,47,62,72,93,108,136,156,199,

%T 226,279,321,398,452,555,630,767,873,1051,1188,1433,1618,1930,2185,

%U 2595,2921,3458,3891,4580,5155,6036,6776,7926,8883,10324,11577,13421,15014

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

%e The a(2) = 1 through a(9) = 9 partitions:

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

%e        (21)  (211)  (311)   (51)     (43)      (71)       (63)

%e                     (2111)  (3111)   (421)     (431)      (621)

%e                             (21111)  (511)     (4211)     (711)

%e                                      (31111)   (5111)     (4311)

%e                                      (211111)  (311111)   (42111)

%e                                                (2111111)  (51111)

%e                                                           (3111111)

%e                                                           (21111111)

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

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

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

%Y Ranked by A331915 = positions of one in A257994.

%Y For a unique composite part we have A379302 (strict A379303), ranks A379301.

%Y The strict case is A379305.

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

%Y Considering 1 prime gives A379314 (strict A379315), 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, A204389, A302540, A330944.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 27 2024