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A083710
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Number of integer partitions of n with a part dividing all the other parts.
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37
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1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 37, 43, 70, 78, 114, 143, 196, 232, 330, 386, 530, 641, 836, 1003, 1340, 1581, 2037, 2461, 3127, 3719, 4746, 5605, 7038, 8394, 10376, 12327, 15272, 17978, 22024, 26095, 31730, 37339, 45333, 53175, 64100, 75340, 90138
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OFFSET
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0,3
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COMMENTS
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Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part." - Joerg Arndt, Jun 08 2009
The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5) - 1 = 6, a(7) = A000041(7) - 3 = 12. - Vladeta Jovovic, Jun 17 2003
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REFERENCES
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L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.
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LINKS
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FORMULA
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Comment from Joerg Arndt, Jun 08 2009: Sequence has g.f. 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n.
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EXAMPLE
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The a(1) = 1 through a(7) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (41) (33) (61)
(111) (31) (221) (42) (331)
(211) (311) (51) (421)
(1111) (2111) (222) (511)
(11111) (321) (2221)
(411) (3211)
(2211) (4111)
(3111) (22111)
(21111) (31111)
(111111) (211111)
(1111111)
(End)
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MAPLE
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with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-0 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], And@@IntegerQ/@(#/Min@@#)&]], {n, 0, 30}] (* Gus Wiseman, Apr 18 2021 *)
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CROSSREFS
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The version for "divisible by" instead of "dividing" is A130689.
The case where there is also a part divisible by all the others is A130714.
The complement of these partitions is counted by A338470.
The Heinz numbers of these partitions are dense, complement of A342193.
The case where there is also no part divisible by all the others is A343345.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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