A379314 Number of integer partitions of n with a unique 1 or prime part.

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, 103, 69, 140, 94, 185, 132, 239, 176, 323, 232, 417, 320, 536, 414, 704, 544, 900, 721, 1145, 936, 1481, 1198, 1867, 1571, 2363, 2001, 3003, 2550, 3768, 3275, 4712

Offset: 0

Author

Gus Wiseman, Dec 28 2024

Keywords

• nonn

Examples

			The a(10) = 3 through a(15) = 14 partitions:
  (8,2)    (11)     (9,3)    (13)       (9,5)      (8,7)
  (9,1)    (6,5)    (10,2)   (7,6)      (12,2)     (10,5)
  (4,4,2)  (7,4)    (6,4,2)  (8,5)      (6,6,2)    (11,4)
           (8,3)             (10,3)     (8,4,2)    (12,3)
           (9,2)             (12,1)     (9,4,1)    (14,1)
           (10,1)            (5,4,4)    (4,4,4,2)  (6,5,4)
           (4,4,3)           (6,4,3)               (6,6,3)
           (6,4,1)           (6,6,1)               (7,4,4)
                             (8,4,1)               (8,4,3)
                             (4,4,4,1)             (8,6,1)
                                                   (9,4,2)
                                                   (10,4,1)
                                                   (4,4,4,3)
                                                   (6,4,4,1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

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

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

Ranked by A379312 = positions of 1 in A379311.

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

The strict case is A379315.

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

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

A000040 lists the prime numbers, differences A001223.

A000041 counts integer partitions, strict A000009.

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

A376682 gives k-th differences of old primes.

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

Programs

Mathematica

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

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