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 A379304 #5 Dec 27 2024 18:08:15
%S A379304 0,0,1,2,2,3,4,6,7,9,11,17,20,26,31,41,47,62,72,93,108,136,156,199,
%T A379304 226,279,321,398,452,555,630,767,873,1051,1188,1433,1618,1930,2185,
%U A379304 2595,2921,3458,3891,4580,5155,6036,6776,7926,8883,10324,11577,13421,15014
%N A379304 Number of integer partitions of n with a unique prime part.
%e A379304 The a(2) = 1 through a(9) = 9 partitions:
%e A379304 (2) (3) (31) (5) (42) (7) (62) (54)
%e A379304 (21) (211) (311) (51) (43) (71) (63)
%e A379304 (2111) (3111) (421) (431) (621)
%e A379304 (21111) (511) (4211) (711)
%e A379304 (31111) (5111) (4311)
%e A379304 (211111) (311111) (42111)
%e A379304 (2111111) (51111)
%e A379304 (3111111)
%e A379304 (21111111)
%t A379304 Table[Length[Select[IntegerPartitions[n],Count[#,_?PrimeQ]==1&]],{n,0,30}]
%Y A379304 For all prime parts we have A000607 (strict A000586), ranks A076610.
%Y A379304 For no prime parts we have A002095 (strict A096258), ranks A320628.
%Y A379304 Ranked by A331915 = positions of one in A257994.
%Y A379304 For a unique composite part we have A379302 (strict A379303), ranks A379301.
%Y A379304 The strict case is A379305.
%Y A379304 For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
%Y A379304 Considering 1 prime gives A379314 (strict A379315), ranks A379312.
%Y A379304 A000040 lists the prime numbers, differences A001223.
%Y A379304 A000041 counts integer partitions, strict A000009.
%Y A379304 A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
%Y A379304 A095195 gives k-th differences of prime numbers.
%Y A379304 Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.
%K A379304 nonn
%O A379304 0,4
%A A379304 _Gus Wiseman_, Dec 27 2024