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A168659
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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.
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34
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1, 1, 2, 2, 3, 3, 6, 6, 8, 9, 14, 16, 22, 25, 33, 39, 51, 60, 79, 92, 116, 137, 174, 204, 254, 300, 368, 435, 530, 625, 760, 896, 1076, 1267, 1518, 1780, 2121, 2484, 2946, 3444, 4070, 4749, 5594, 6514, 7637, 8879, 10384, 12043, 14040, 16255
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OFFSET
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1,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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
The a(1) = 1 through a(10) = 9 partitions of the first type:
1 11 21 22 311 321 322 332 333 4222
111 1111 2111 2211 331 2222 4221 4321
11111 111111 2221 4211 4311 4411
4111 221111 51111 52111
211111 311111 222111 222211
1111111 11111111 321111 322111
21111111 331111
111111111 22111111
1111111111
The a(1) = 1 through a(11) = 14 partitions of the second type (A=10, B=11):
1 2 3 4 5 6 7 8 9 A B
21 22 41 42 43 44 63 64 65
311 321 61 62 81 82 83
322 332 333 622 A1
331 611 621 631 632
4111 4211 4221 4222 641
4311 4321 911
51111 4411 4322
52111 4331
4421
8111
52211
53111
611111
(End)
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MAPLE
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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
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Divisible[Length[#], Max[#]]&]], {n, 30}] (* Gus Wiseman, Feb 08 2021 *)
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CROSSREFS
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Note: A-numbers of Heinz-number sequences are in parentheses below.
If all parts (not just the greatest) are divisors we get A340693 (A340606).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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