cp's OEIS Frontend

A343347 Number of strict integer partitions of n with a part divisible by all the others.

%I A343347 #10 Apr 19 2021 08:05:39
%S A343347 1,1,1,2,2,2,3,3,3,4,4,4,6,5,4,6,6,6,8,7,7,10,9,9,12,10,8,11,11,10,14,
%T A343347 13,11,13,12,15,20,17,15,19,19,19,22,18,17,23,22,22,28,25,24,31,28,26,
%U A343347 32,32,30,34,32,29,37,33,27,36,33,34,44,38,36,45,45
%N A343347 Number of strict integer partitions of n with a part divisible by all the others.
%C A343347 Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.
%H A343347 Andrew Howroyd, <a href="/A343347/b343347.txt">Table of n, a(n) for n = 0..1000</a>
%F A343347 G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - _Andrew Howroyd_, Apr 17 2021
%e A343347 The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
%e A343347   1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
%e A343347         21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
%e A343347                     51  421  71  81   91   632  93    841  D1   C3
%e A343347                                  621  631  821  A2    931  842  E1
%e A343347                                                 B1    A21       C21
%e A343347                                                 6321            8421
%t A343347 Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%o A343347 (PARI) seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ _Andrew Howroyd_, Apr 17 2021
%Y A343347 The dual version is A097986 (non-strict: A083710).
%Y A343347 The non-strict version is A130689 (Heinz numbers: complement of A343337).
%Y A343347 The strict complement is counted by A343377.
%Y A343347 The case with smallest part divisible by all the others is A343378.
%Y A343347 The case with smallest part not divisible by all the others is A343380.
%Y A343347 A000005 counts divisors.
%Y A343347 A000009 counts strict partitions.
%Y A343347 A000070 counts partitions with a selected part.
%Y A343347 A015723 counts strict partitions with a selected part.
%Y A343347 A018818 counts partitions into divisors (strict: A033630).
%Y A343347 A167865 counts strict chains of divisors > 1 summing to n.
%Y A343347 A339564 counts factorizations with a selected factor.
%Y A343347 Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.
%K A343347 nonn
%O A343347 0,4
%A A343347 _Gus Wiseman_, Apr 16 2021