A098965 - OEIS

%I #19 Nov 20 2024 08:46:52

%S 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 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 120,1,196,1,160,42,201,10,259,1,258,50,292,1,407,1,357,81,431,8,548,1,577

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

%C If n > 0, we can assume this part is the smallest. - _Gus Wiseman_, Apr 18 2021

%H Alois P. Heinz, <a href="/A098965/b098965.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, d<n} A025147(d-1).

%F G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).

%t 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 Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* _Gus Wiseman_, Apr 18 2021 *)

%Y The non-strict version with 1's allowed is A083710.

%Y The non-strict version is A083711.

%Y The version with 1's allowed is A097986.

%Y The Heinz numbers of these partitions are the odd terms of A339563.

%Y The non-strict dual is A339619.

%Y The strict complement is counted by A341450.

%Y A000005 counts divisors.

%Y A000041 counts partitions.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

%Y A018818 counts partitions into divisors (strict: A033630).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.

%K easy,nonn

%O 1,6

%A _Vladeta Jovovic_, Oct 23 2004

%E More terms from _Robert G. Wilson v_, Nov 01 2004

%E Name shortened by _Gus Wiseman_, Apr 23 2021