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A332977
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Triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n; n>=0, 1<=k<=A011782(n).
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6
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1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 25, 27, 11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125, 13, 22, 28, 33, 40, 42, 48, 55, 60, 63, 64, 70, 72, 77, 90, 98, 100, 105, 108, 121, 135, 147, 150, 162, 175, 225, 243, 245, 250, 343, 375, 625
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OFFSET
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0,2
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COMMENTS
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Integer m > 0 is listed in row n if the index of the largest prime factor of m (or 0 for empty prime factor set) plus the cardinality of the other prime factors of m (counted with multiplicity) equals n.
Row n+k-1 contains prime(n)^k (for all n, k >= 1).
The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1, 2, 3, 4, 5, 6, 10, ... and inverse permutation A332990.
This is a variant with sorted rows of A005940 (offset differs) or A163511.
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LINKS
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EXAMPLE
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Triangle T(n,k) begins:
1;
2;
3, 4;
5, 6, 8, 9;
7, 10, 12, 15, 16, 18, 25, 27;
11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125;
...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [1], sort([seq(map(x-> x*
ithprime(j), b(n-`if`(i=0, j, 1), j))[], j=1..`if`(i=0, n, i))]))
end:
T:= n-> b(n, 0)[]:
seq(T(n), n=0..7);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, {1}, Sort[Flatten[Table[#*
Prime[j]& /@ b[n-If[i == 0, j, 1], j], {j, 1, If[i == 0, n, i]}]]]];
T[n_] := b[n, 0];
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CROSSREFS
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Last elements of rows give A332979.
Product of row elements give A252738.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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