cp's OEIS Frontend

A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

%I A380411 #9 Jan 29 2025 13:08:44
%S A380411 1,0,0,0,0,0,0,1,4,8,14,23,39,58,85,121,168,228,308,404,533,691,892,
%T A380411 1136,1449,1820,2291,2857,3553,4387,5418,6646,8144,9931,12086,14649,
%U A380411 17733,21379,25747,30905,37049,44282,52863,62936,74841,88792,105202,124387
%N A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.
%e A380411 The partition y = (4,3,2) has product of parts 4*3*2 = 24 and sum of corresponding primes 7+5+3 = 15, so y is counted under a(9).
%e A380411 The a(0) = 1 through a(10) = 14 partitions:
%e A380411   ()  .  .  .  .  .  .  (322)  (44)    (54)     (55)
%e A380411                                (332)   (333)    (64)
%e A380411                                (422)   (432)    (433)
%e A380411                                (2222)  (522)    (442)
%e A380411                                        (3222)   (532)
%e A380411                                        (3321)   (622)
%e A380411                                        (4221)   (3322)
%e A380411                                        (22221)  (3331)
%e A380411                                                 (4222)
%e A380411                                                 (4321)
%e A380411                                                 (5221)
%e A380411                                                 (22222)
%e A380411                                                 (32221)
%e A380411                                                 (33211)
%t A380411 Table[Length[Select[IntegerPartitions[n],Times@@#>Plus@@Prime/@#&]],{n,0,30}]
%Y A380411 For parts instead of primes on the RHS we have A114324.
%Y A380411 The version for divisibility instead of inequality is A330954.
%Y A380411 The version for equality is A331383, ranks A331384.
%Y A380411 These partitions are ranked by A380410.
%Y A380411 A000040 lists the primes, differences A001223.
%Y A380411 A000041 counts integer partitions, strict A000009.
%Y A380411 A001414 gives sum of prime factors.
%Y A380411 A003963 gives product of prime indices
%Y A380411 A379666 counts partitions by sum and product.
%Y A380411 Counting and ranking multisets by comparing sum and product:
%Y A380411 - same: A001055, ranks A301987
%Y A380411 - multiple: A057567, ranks A326155
%Y A380411 - divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
%Y A380411 - greater than: A096276 shifted right, ranks A325038
%Y A380411 - greater or equal: A096276, ranks A325044
%Y A380411 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380411 - less or equal: A319005, ranks A379721, see A025147
%Y A380411 - different: A379736, ranks A379722, see A111133
%Y A380411 Cf. A175508, A318950, A319000, A319916, A326152, A379320, A379734, A380218, A380219, A380344.
%K A380411 nonn
%O A380411 0,9
%A A380411 _Gus Wiseman_, Jan 26 2025