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A327540
Number of factorizations of A327534(n), the n-th number that is 1, prime, or whose prime indices are relatively prime, into numbers > 1 satisfying the same conditions.
0
1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 1, 7, 2, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 7, 1, 4, 1, 4, 3, 2, 1, 12, 3, 2, 4, 1, 4, 2, 7, 2, 1, 11, 1, 2, 11, 5, 1, 4, 2, 5, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 2, 1, 9, 2, 2, 7, 1, 9, 4, 2, 2, 2, 19, 1
OFFSET
1,4
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 that are 1, prime, or whose prime indices are relatively prime are A327534. The number of divisors of n satisfying the same conditions is A327536(n).
EXAMPLE
The a(74) = 9 factorizations of 84 together with the corresponding multiset partitions of {1,1,2,4}:
(2*2*3*7) {{1},{1},{2},{4}}
(2*3*14) {{1},{2},{1,4}}
(2*6*7) {{1},{1,2},{4}}
(2*42) {{1},{1,2,4}}
(3*4*7) {{2},{1,1},{4}}
(3*28) {{2},{1,1,4}}
(6*14) {{1,2},{1,4}}
(7*12) {{4},{1,1,2}}
(84) {{1,1,2,4}}
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||PrimeQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&];
Table[Length[facsusing[Rest[y], n]], {n, y}]
CROSSREFS
See link for additional cross-references.
Sequence in context: A087467 A231568 A327525 * A227687 A128118 A205696
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2019
STATUS
approved