cp's OEIS Frontend

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

%I A379315 #10 Dec 29 2024 00:42:00
%S A379315 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,
%T A379315 38,62,48,76,68,93,80,119,105,147,137,175,166,226,208,267,263,326,322,
%U A379315 407,391,481,492,586,591,714,714,849,884,1020,1050,1232,1263
%N A379315 Number of strict integer partitions of n with a unique 1 or prime part.
%C A379315 The "old" primes are listed by A008578.
%H A379315 Andrew Howroyd, <a href="/A379315/b379315.txt">Table of n, a(n) for n = 0..1000</a>
%e A379315 The a(10) = 2 through a(15) = 10 partitions:
%e A379315   (8,2)  (11)     (9,3)    (13)     (9,5)    (8,7)
%e A379315   (9,1)  (6,5)    (10,2)   (7,6)    (12,2)   (10,5)
%e A379315          (7,4)    (6,4,2)  (8,5)    (8,4,2)  (11,4)
%e A379315          (8,3)             (10,3)   (9,4,1)  (12,3)
%e A379315          (9,2)             (12,1)            (14,1)
%e A379315          (10,1)            (6,4,3)           (6,5,4)
%e A379315          (6,4,1)           (8,4,1)           (8,4,3)
%e A379315                                              (8,6,1)
%e A379315                                              (9,4,2)
%e A379315                                              (10,4,1)
%t A379315 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
%o A379315 (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 A379315 For all prime parts we have A000586, non-strict A000607 (ranks A076610).
%Y A379315 For no prime parts we have A096258, non-strict A002095 (ranks A320628).
%Y A379315 For a unique composite part we have A379303, non-strict A379302 (ranks A379301).
%Y A379315 Considering 1 nonprime gives A379305, non-strict A379304 (ranks A331915).
%Y A379315 For squarefree instead of old prime we have A379309, non-strict A379308 (ranks A379316).
%Y A379315 Ranked by A379312 /\ A005117 = squarefree positions of 1 in A379311.
%Y A379315 The non-strict version is A379314.
%Y A379315 A000040 lists the prime numbers, differences A001223.
%Y A379315 A000041 counts integer partitions, strict A000009.
%Y A379315 A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
%Y A379315 A376682 gives k-th differences of old primes.
%Y A379315 Cf. A000070, A023895, A034891, A036497, A038348, A095195, A175804, A204389, A257994, A302540.
%K A379315 nonn
%O A379315 0,6
%A A379315 _Gus Wiseman_, Dec 28 2024