This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379314 #11 Dec 29 2024 00:42:05
%S A379314 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 A379314 103,69,140,94,185,132,239,176,323,232,417,320,536,414,704,544,900,
%U A379314 721,1145,936,1481,1198,1867,1571,2363,2001,3003,2550,3768,3275,4712
%N A379314 Number of integer partitions of n with a unique 1 or prime part.
%H A379314 Andrew Howroyd, <a href="/A379314/b379314.txt">Table of n, a(n) for n = 0..1000</a>
%e A379314 The a(10) = 3 through a(15) = 14 partitions:
%e A379314 (8,2) (11) (9,3) (13) (9,5) (8,7)
%e A379314 (9,1) (6,5) (10,2) (7,6) (12,2) (10,5)
%e A379314 (4,4,2) (7,4) (6,4,2) (8,5) (6,6,2) (11,4)
%e A379314 (8,3) (10,3) (8,4,2) (12,3)
%e A379314 (9,2) (12,1) (9,4,1) (14,1)
%e A379314 (10,1) (5,4,4) (4,4,4,2) (6,5,4)
%e A379314 (4,4,3) (6,4,3) (6,6,3)
%e A379314 (6,4,1) (6,6,1) (7,4,4)
%e A379314 (8,4,1) (8,4,3)
%e A379314 (4,4,4,1) (8,6,1)
%e A379314 (9,4,2)
%e A379314 (10,4,1)
%e A379314 (4,4,4,3)
%e A379314 (6,4,4,1)
%t A379314 Table[Length[Select[IntegerPartitions[n],Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
%o A379314 (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 A379314 For all prime parts we have A000607 (strict A000586), ranks A076610.
%Y A379314 For no prime parts we have A002095 (strict A096258), ranks A320628.
%Y A379314 Ranked by A379312 = positions of 1 in A379311.
%Y A379314 For a unique composite part we have A379302 (strict A379303), ranks A379301.
%Y A379314 The strict case is A379315.
%Y A379314 For squarefree instead of old prime we have A379308 (strict A379309), ranks A379316.
%Y A379314 Considering 1 nonprime gives A379304 (strict A379305), ranks A331915.
%Y A379314 A000040 lists the prime numbers, differences A001223.
%Y A379314 A000041 counts integer partitions, strict A000009.
%Y A379314 A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
%Y A379314 A376682 gives k-th differences of old primes.
%Y A379314 Cf. A000070, A023895, A034891, A036497, A095195, A175804, A204389, A257994, A302540, A330944.
%K A379314 nonn
%O A379314 0,6
%A A379314 _Gus Wiseman_, Dec 28 2024