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A327533
Number of factorizations of the n-th number that is 1 or whose prime indices are relatively prime A289509(n - 1) into numbers > 1 satisfying the same conditions.
0
1, 1, 2, 1, 3, 1, 2, 1, 1, 5, 1, 2, 1, 4, 1, 2, 2, 7, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 7, 1, 1, 2, 1, 1, 4, 1, 5, 1, 11, 2, 2, 1, 2, 6, 1, 1, 2, 1, 1, 7, 1, 3, 1, 1, 4, 3, 2, 1, 1, 1, 12, 1, 1, 3, 2, 4, 1, 1, 3, 2, 7, 1, 2, 1, 1, 10, 1, 1, 2, 1, 15, 1, 5, 1, 1, 4
OFFSET
1,3
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Numbers whose prime indices are relatively prime are A289509.
EXAMPLE
The a(76) = 10 factorizations of 120 into elements of A289509 together with the corresponding multiset partitions of {1,1,1,2,3}:
(2*2*2*15) {{1},{1},{1},{2,3}}
(2*2*30) {{1},{1},{1,2,3}}
(2*4*15) {{1},{1,1},{2,3}}
(2*6*10) {{1},{1,2},{1,3}}
(2*60) {{1},{1,1,2,3}}
(4*30) {{1,1},{1,2,3}}
(6*20) {{1,2},{1,1,3}}
(8*15) {{1,1,1},{2,3}}
(10*12) {{1,3},{1,1,2}}
(120) {{1,1,1,2,3}}
MATHEMATICA
nn=100;
facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];
y=Select[Range[nn], #==1||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
Table[Length[facsusing[Rest[y], n]], {n, y}]
CROSSREFS
See link for additional cross-references.
Sequence in context: A228098 A366833 A353161 * A327518 A174532 A089242
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2019
STATUS
approved