A379314 - OEIS

%I #11 Dec 29 2024 00:42:05

%S 0,1,1,1,0,2,1,3,1,4,3,8,3,10,6,14,8,22,12,30,18,40,26,58,33,76,53,

%T 103,69,140,94,185,132,239,176,323,232,417,320,536,414,704,544,900,

%U 721,1145,936,1481,1198,1867,1571,2363,2001,3003,2550,3768,3275,4712

%N Number of integer partitions of n with a unique 1 or prime part.

%H Andrew Howroyd, <a href="/A379314/b379314.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(10) = 3 through a(15) = 14 partitions:

%e   (8,2)    (11)     (9,3)    (13)       (9,5)      (8,7)

%e   (9,1)    (6,5)    (10,2)   (7,6)      (12,2)     (10,5)

%e   (4,4,2)  (7,4)    (6,4,2)  (8,5)      (6,6,2)    (11,4)

%e            (8,3)             (10,3)     (8,4,2)    (12,3)

%e            (9,2)             (12,1)     (9,4,1)    (14,1)

%e            (10,1)            (5,4,4)    (4,4,4,2)  (6,5,4)

%e            (4,4,3)           (6,4,3)               (6,6,3)

%e            (6,4,1)           (6,6,1)               (7,4,4)

%e                              (8,4,1)               (8,4,3)

%e                              (4,4,4,1)             (8,6,1)

%e                                                    (9,4,2)

%e                                                    (10,4,1)

%e                                                    (4,4,4,3)

%e                                                    (6,4,4,1)

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

%o (PARI) seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k))/prod(k=4, n, 1 - if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ _Andrew Howroyd_, Dec 28 2024

%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 A379312 = positions of 1 in A379311.

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

%Y The strict case is A379315.

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

%Y Considering 1 nonprime gives A379304 (strict A379305), ranks A331915.

%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 A376682 gives k-th differences of old primes.

%Y Cf. A000070, A023895, A034891, A036497, A095195, A175804, A204389, A257994, A302540, A330944.

%K nonn,new

%O 0,6

%A _Gus Wiseman_, Dec 28 2024