This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A097986 #33 Jul 06 2025 09:00:11
%S A097986 1,1,2,2,2,4,3,5,5,7,6,12,9,13,15,20,18,28,26,37,39,47,49,71,68,85,94,
%T A097986 117,120,159,160,201,216,257,277,348,357,430,470,562,592,720,758,901,
%U A097986 981,1134,1220,1457,1542,1798,1952,2250,2419,2819,3023,3482,3773,4291
%N A097986 Number of strict integer partitions of n with a part dividing all the other parts.
%C A097986 If n > 0, we can assume such a part is the smallest. - _Gus Wiseman_, Apr 23 2021
%C A097986 Also the number of uniform (constant multiplicity) partitions of n containing 1, ranked by A367586. The strict case is A096765. The version without 1 is A329436. - _Gus Wiseman_, Dec 01 2023
%H A097986 Alois P. Heinz, <a href="/A097986/b097986.txt">Table of n, a(n) for n = 1..10000</a> (first 500 terms from John Tyler Rascoe)
%F A097986 a(n) = Sum_{d|n} A025147(d-1).
%F A097986 G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).
%F A097986 a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jul 06 2025
%e A097986 From _Gus Wiseman_, Dec 01 2023: (Start)
%e A097986 The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:
%e A097986 (1) (2) (3) (4) (5) (6) (7) (8)
%e A097986 (2,1) (3,1) (4,1) (4,2) (6,1) (6,2)
%e A097986 (5,1) (4,2,1) (7,1)
%e A097986 (3,2,1) (4,3,1)
%e A097986 (5,2,1)
%e A097986 The a(1) = 1 through a(8) = 5 uniform partitions containing 1:
%e A097986 (1) (11) (21) (31) (41) (51) (61) (71)
%e A097986 (111) (1111) (11111) (321) (421) (431)
%e A097986 (2211) (1111111) (521)
%e A097986 (111111) (3311)
%e A097986 (11111111)
%e A097986 (End)
%t A097986 Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (* _Robert G. Wilson v_, Nov 01 2004 *)
%t A097986 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x,#}]&]], {n,0,30}] (* _Gus Wiseman_, Apr 23 2021 *)
%o A097986 (PARI)
%o A097986 A_x(N) = {my(x='x+O('x^N)); Vec(sum(k=1,N,x^k*prod(i=2,N-k, (1+x^(k*i)))))}
%o A097986 A_x(50) \\ _John Tyler Rascoe_, Nov 19 2024
%Y A097986 The non-strict version is A083710.
%Y A097986 The case with no 1's is A098965.
%Y A097986 The Heinz numbers of these partitions are A339563.
%Y A097986 The strict complement is counted by A341450.
%Y A097986 The version for "divisible by" instead of "dividing" is A343347.
%Y A097986 The case where there is also a part divisible by all the others is A343378.
%Y A097986 The case where there is no part divisible by all the others is A343381.
%Y A097986 A000005 counts divisors.
%Y A097986 A000009 counts strict partitions.
%Y A097986 A000070 counts partitions with a selected part.
%Y A097986 A006128 counts partitions with a selected position.
%Y A097986 A015723 counts strict partitions with a selected part.
%Y A097986 A018818 counts partitions into divisors (strict: A033630).
%Y A097986 A167865 counts strict chains of divisors > 1 summing to n.
%Y A097986 Cf. A083711, A098743, A130689, A200745, A264401, A338470, A343377, A343379, A343380.
%Y A097986 Cf. A023645, A025147, A047966, A072774, A096765, A329436, A367586.
%K A097986 easy,nonn
%O A097986 1,3
%A A097986 _Vladeta Jovovic_, Oct 23 2004
%E A097986 More terms from _Robert G. Wilson v_, Nov 01 2004
%E A097986 Name shortened by _Gus Wiseman_, Apr 23 2021