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 A168659 #43 Feb 07 2025 00:51:21
%S A168659 1,1,2,2,3,3,6,6,8,9,14,16,22,25,33,39,51,60,79,92,116,137,174,204,
%T A168659 254,300,368,435,530,625,760,896,1076,1267,1518,1780,2121,2484,2946,
%U A168659 3444,4070,4749,5594,6514,7637,8879,10384,12043,14040,16255
%N A168659 Number of partitions of n such that the number of parts is divisible by the greatest part. Also number of partitions of n such that the greatest part is divisible by the number of parts.
%H A168659 Vaclav Kotesovec, <a href="/A168659/b168659.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..301 from Vladeta Jovovic corrected by N. J. A. Sloane, Oct 05 2010, terms 302..1000 from Seiichi Manyama)
%F A168659 G.f.: Sum_{i>=1} Sum_{j>=1} x^((i+1)*j-1) * Product_{k=1..j-1} (1-x^(i*j+k-1))/(1-x^k). - _Seiichi Manyama_, Jan 24 2022
%F A168659 a(n) ~ c * exp(Pi*sqrt(2*n/3)) / n^(3/2), where c = 0.04628003... - _Vaclav Kotesovec_, Nov 16 2024
%e A168659 a(5)=3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,1,3] the number of parts is divisible by the greatest part; not true for the partitions [1,2,2],[2,3], [1,4], and [5]. - _Emeric Deutsch_, Dec 04 2009
%e A168659 From _Gus Wiseman_, Feb 08 2021: (Start)
%e A168659 The a(1) = 1 through a(10) = 9 partitions of the first type:
%e A168659 1 11 21 22 311 321 322 332 333 4222
%e A168659 111 1111 2111 2211 331 2222 4221 4321
%e A168659 11111 111111 2221 4211 4311 4411
%e A168659 4111 221111 51111 52111
%e A168659 211111 311111 222111 222211
%e A168659 1111111 11111111 321111 322111
%e A168659 21111111 331111
%e A168659 111111111 22111111
%e A168659 1111111111
%e A168659 The a(1) = 1 through a(11) = 14 partitions of the second type (A=10, B=11):
%e A168659 1 2 3 4 5 6 7 8 9 A B
%e A168659 21 22 41 42 43 44 63 64 65
%e A168659 311 321 61 62 81 82 83
%e A168659 322 332 333 622 A1
%e A168659 331 611 621 631 632
%e A168659 4111 4211 4221 4222 641
%e A168659 4311 4321 911
%e A168659 51111 4411 4322
%e A168659 52111 4331
%e A168659 4421
%e A168659 8111
%e A168659 52211
%e A168659 53111
%e A168659 611111
%e A168659 (End)
%p A168659 a := proc (n) local pn, ct, j: with(combinat): pn := partition(n): ct := 0: for j to numbpart(n) do if `mod`(nops(pn[j]), max(seq(pn[j][i], i = 1 .. nops(pn[j])))) = 0 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 50); # _Emeric Deutsch_, Dec 04 2009
%t A168659 Table[Length[Select[IntegerPartitions[n],Divisible[Length[#],Max[#]]&]],{n,30}] (* _Gus Wiseman_, Feb 08 2021 *)
%t A168659 nmax = 100; s = 0; Do[s += Normal[Series[Sum[x^((m+1)*k - 1) * Product[(1 - x^(m*k + j - 1))/(1 - x^j), {j, 1, k-1}], {k, 1, (1 + nmax)/(1 + m) + 1}], {x, 0, nmax}]], {m, 1, nmax}]; Rest[CoefficientList[s, x]] (* _Vaclav Kotesovec_, Oct 18 2024 *)
%Y A168659 Cf. A168656, A168657, A079501, A168655.
%Y A168659 Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y A168659 The case of equality is A047993 (A106529).
%Y A168659 The Heinz numbers of these partitions are A340609/A340610.
%Y A168659 If all parts (not just the greatest) are divisors we get A340693 (A340606).
%Y A168659 The strict case in the second interpretation is A340828 (A340856).
%Y A168659 A006141 = partitions whose length equals their minimum (A324522).
%Y A168659 A067538 = partitions whose length/max divides their sum (A316413/A326836).
%Y A168659 A200750 = partitions with length coprime to maximum (A340608).
%Y A168659 Cf. A003114, A039900, A064173, A064174, A143773, A326837, A340653, A340830, A340851, A340853.
%Y A168659 Row sums of A350879.
%K A168659 nonn
%O A168659 1,3
%A A168659 _Vladeta Jovovic_, Dec 02 2009
%E A168659 Extended by _Emeric Deutsch_, Dec 04 2009