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A332977 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).

%I A332977 #37 Apr 10 2021 15:27:11
%S A332977 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,
%T A332977 49,50,54,75,81,125,13,22,28,33,40,42,48,55,60,63,64,70,72,77,90,98,
%U A332977 100,105,108,121,135,147,150,162,175,225,243,245,250,343,375,625
%N A332977 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).
%C A332977 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.
%C A332977 Row n+k-1 contains prime(n)^k (for all n, k >= 1).
%C A332977 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.
%C A332977 This is a variant with sorted rows of A005940 (offset differs) or A163511.
%H A332977 Alois P. Heinz, <a href="/A332977/b332977.txt">Rows n = 0..16, flattened</a>
%H A332977 Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%H A332977 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A332977 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A332977 Triangle T(n,k) begins:
%e A332977    1;
%e A332977    2;
%e A332977    3,  4;
%e A332977    5,  6,  8,  9;
%e A332977    7, 10, 12, 15, 16, 18, 25, 27;
%e A332977   11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125;
%e A332977   ...
%p A332977 b:= proc(n, i) option remember; `if`(n=0, [1], sort([seq(map(x-> x*
%p A332977       ithprime(j), b(n-`if`(i=0, j, 1), j))[], j=1..`if`(i=0, n, i))]))
%p A332977     end:
%p A332977 T:= n-> b(n, 0)[]:
%p A332977 seq(T(n), n=0..7);
%t A332977 b[n_, i_] := b[n, i] = If[n == 0, {1}, Sort[Flatten[Table[#*
%t A332977     Prime[j]& /@ b[n-If[i == 0, j, 1], j], {j, 1, If[i == 0, n, i]}]]]];
%t A332977 T[n_] := b[n, 0];
%t A332977 T /@ Range[0, 7] // Flatten (* _Jean-François Alcover_, Mar 30 2021, after _Alois P. Heinz_ *)
%Y A332977 Columns k=1-2 give: A008578(n+1), A100484(n-1) for n>1.
%Y A332977 Last elements of rows give A332979.
%Y A332977 Row sums give A252737.
%Y A332977 Product of row elements give A252738.
%Y A332977 Row lengths give A011782.
%Y A332977 Cf. A000027, A000040, A000720 (pi), A001222 (Omega), A006530 (GPF), A060576 ([n<>1]), A061395 (pi(gpf(n))), A215366, A332990.
%Y A332977 Cf. A005940, A163511.
%K A332977 nonn,tabf
%O A332977 0,2
%A A332977 _Alois P. Heinz_, Mar 04 2020