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A343347 Number of strict integer partitions of n with a part divisible by all the others. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.
LINKS
FORMULA
G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021
EXAMPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
PROG
(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
CROSSREFS
The dual version is A097986 (non-strict: A083710).
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.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Sequence in context: A062276 A350327 A053264 * A079440 A192262 A026414
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 16 2021
STATUS
approved

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Last modified March 19 07:14 EDT 2024. Contains 370954 sequences. (Running on oeis4.)