A379315 Number of strict integer partitions of n with a unique 1 or prime part.

0, 1, 1, 1, 0, 2, 1, 3, 1, 3, 2, 7, 3, 7, 4, 10, 7, 15, 7, 17, 13, 23, 16, 31, 20, 37, 31, 48, 38, 62, 48, 76, 68, 93, 80, 119, 105, 147, 137, 175, 166, 226, 208, 267, 263, 326, 322, 407, 391, 481, 492, 586, 591, 714, 714, 849, 884, 1020, 1050, 1232, 1263

Offset: 0

Author

Gus Wiseman, Dec 28 2024

Keywords

• nonn

Comments

The "old" primes are listed by A008578.

Examples

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

Links

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

Crossrefs

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

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

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

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

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

Ranked by A379312 /\ A005117 = squarefree positions of 1 in A379311.

The non-strict version is A379314.

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, A038348, A095195, A175804, A204389, A257994, A302540.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&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