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A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

%I A098965 #19 Nov 20 2024 08:46:52
%S A098965 0,1,1,1,1,2,1,2,2,2,1,5,1,3,3,5,1,7,1,8,4,6,1,15,2,9,5,14,1,22,1,20,
%T A098965 7,18,4,36,1,26,10,40,1,51,1,48,18,49,1,86,3,73,19,86,1,117,7,120,27,
%U A098965 120,1,196,1,160,42,201,10,259,1,258,50,292,1,407,1,357,81,431,8,548,1,577
%N A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.
%C A098965 If n > 0, we can assume this part is the smallest. - _Gus Wiseman_, Apr 18 2021
%H A098965 Alois P. Heinz, <a href="/A098965/b098965.txt">Table of n, a(n) for n = 1..10000</a>
%F A098965 a(n) = Sum_{d|n, d<n} A025147(d-1).
%F A098965 G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).
%t A098965 Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 92}], {k, 2, 92}]], x], {2, 81}] (* _Robert G. Wilson v_, Nov 01 2004 *)
%t A098965 Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* _Gus Wiseman_, Apr 18 2021 *)
%Y A098965 The non-strict version with 1's allowed is A083710.
%Y A098965 The non-strict version is A083711.
%Y A098965 The version with 1's allowed is A097986.
%Y A098965 The Heinz numbers of these partitions are the odd terms of A339563.
%Y A098965 The non-strict dual is A339619.
%Y A098965 The strict complement is counted by A341450.
%Y A098965 A000005 counts divisors.
%Y A098965 A000041 counts partitions.
%Y A098965 A000070 counts partitions with a selected part.
%Y A098965 A006128 counts partitions with a selected position.
%Y A098965 A015723 counts strict partitions with a selected part.
%Y A098965 A018818 counts partitions into divisors (strict: A033630).
%Y A098965 A167865 counts strict chains of divisors > 1 summing to n.
%Y A098965 Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.
%K A098965 easy,nonn
%O A098965 1,6
%A A098965 _Vladeta Jovovic_, Oct 23 2004
%E A098965 More terms from _Robert G. Wilson v_, Nov 01 2004
%E A098965 Name shortened by _Gus Wiseman_, Apr 23 2021