cp's OEIS Frontend

A379309 Number of strict integer partitions of n with a unique squarefree part.

%I A379309 #13 Feb 21 2025 05:32:13
%S A379309 0,1,1,1,0,2,2,2,0,2,4,4,1,4,7,7,2,6,8,11,4,9,13,17,7,13,20,22,13,20,
%T A379309 29,33,21,29,40,47,27,41,56,64,42,59,77,85,60,74,104,115,83,101,141,
%U A379309 155,113,138,179,206,156,183,236,272,212,239,309,343,282,315
%N A379309 Number of strict integer partitions of n with a unique squarefree part.
%e A379309 The a(9) = 2 through a(15) = 7 partitions:
%e A379309   (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
%e A379309   (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
%e A379309          (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
%e A379309          (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
%e A379309                                        (12,2)   (12,3)
%e A379309                                        (8,4,2)  (8,4,3)
%e A379309                                        (9,4,1)  (9,4,2)
%t A379309 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]
%o A379309 (PARI) lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ _Jinyuan Wang_, Feb 21 2025
%Y A379309 If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).
%Y A379309 If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).
%Y A379309 For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).
%Y A379309 For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).
%Y A379309 The non-strict version is A379308, ranks A379316.
%Y A379309 For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).
%Y A379309 Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.
%Y A379309 A000041 counts integer partitions, strict A000009.
%Y A379309 A005117 lists the squarefree numbers, differences A076259.
%Y A379309 A013929 lists the nonsquarefree numbers, differences A078147.
%Y A379309 A377038 gives k-th differences of squarefree numbers.
%Y A379309 A379310 counts nonsquarefree prime indices.
%Y A379309 Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.
%K A379309 nonn
%O A379309 0,6
%A A379309 _Gus Wiseman_, Dec 27 2024
%E A379309 More terms from _Jinyuan Wang_, Feb 21 2025