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A319299
Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.
2
1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 7, 2, 1, 1, 14, 1, 17, 3, 1, 1, 27, 2, 1, 34, 6, 1, 1, 55, 1, 63, 7, 3, 2, 1, 1, 100, 1, 119, 14, 1, 1, 167, 6, 2, 1, 209, 17, 3, 1, 1, 296, 1, 347, 27, 7, 2, 1, 1, 489, 1, 582, 34, 6, 3, 1, 1, 775, 14, 2, 1, 945, 55, 1, 1, 1254
OFFSET
1,4
LINKS
Robert Israel, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened)
FORMULA
T(n,k) = A000837(n/A027750(n,k)).
EXAMPLE
Triangle begins:
1
1 1
2 1
3 1 1
6 1
7 2 1 1
14 1
17 3 1 1
27 2 1
34 6 1 1
55 1
63 7 3 2 1 1
100 1
119 14 1 1
167 6 2 1
209 17 3 1 1
296 1
347 27 7 2 1 1
489 1
582 34 6 3 1 1
MAPLE
# with table A000837 obtained from that sequence
f:= proc(n) local D, d;
D:= sort(convert(numtheory:-divisors(n), list), `>`);
seq(A000837[d], d=D)
end proc:
map(f, [$1..60]); # Robert Israel, Jul 09 2020
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], GCD@@#==k&]], {n, 20}, {k, Divisors[n]}]
CROSSREFS
A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.
Sequence in context: A191861 A350200 A129761 * A207031 A156248 A324817
KEYWORD
nonn,tabf,look
AUTHOR
Gus Wiseman, Sep 16 2018
STATUS
approved