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1, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 14, 1, 13, 8, 20, 1, 33, 1, 40, 14, 44, 1, 85, 6, 79, 25, 117, 1, 181, 1, 196, 45, 233, 17, 389, 1, 387, 80, 545, 1, 750, 1, 839, 165, 1004, 1, 1516, 12, 1612, 234, 2040, 1, 2766, 48, 3142, 388, 3720, 1, 5295, 1, 5606, 663, 7038, 83, 9194, 1, 10379, 1005
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021
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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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a(n) = Sum_{ d|n, d<n} A000041(d-1).
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EXAMPLE
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The a(6) = 4 through a(12) = 13 partitions:
(6) (7) (8) (9) (10) (11) (12)
(3,3) (4,4) (6,3) (5,5) (6,6)
(4,2) (6,2) (3,3,3) (8,2) (8,4)
(2,2,2) (4,2,2) (4,4,2) (9,3)
(2,2,2,2) (6,2,2) (10,2)
(4,2,2,2) (4,4,4)
(2,2,2,2,2) (6,3,3)
(6,4,2)
(8,2,2)
(3,3,3,3)
(4,4,2,2)
(6,2,2,2)
(4,2,2,2,2)
(2,2,2,2,2,2)
(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)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d, `, a(j)) od: # James A. Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
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MATHEMATICA
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a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
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CROSSREFS
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The complement (except also without 1's) is counted by A338470.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
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,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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