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A343347
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Number of strict integer partitions of n with a part divisible by all the others.
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13
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1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45
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OFFSET
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0,4
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COMMENTS
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Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.
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LINKS
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FORMULA
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G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021
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EXAMPLE
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The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
1 2 3 4 5 6 7 8 9 A B C D E F
21 31 41 42 61 62 63 82 A1 84 C1 C2 A5
51 421 71 81 91 632 93 841 D1 C3
621 631 821 A2 931 842 E1
B1 A21 C21
6321 8421
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
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PROG
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(PARI) seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021
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CROSSREFS
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The non-strict version is A130689 (Heinz numbers: complement of A343337).
The strict complement is counted by A343377.
The case with smallest part divisible by all the others is A343378.
The case with smallest part not divisible by all the others is A343380.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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