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A153734 Triangle T(n,k): T(n,k) gives the A153452(m_k) such that A056239(m_k) = n, [1<=k<=A000041(n)], sorted by m_k, read by rows. Sequence A060240 is this sequence's permutation.

%I A153734 #15 Feb 16 2021 05:57:14
%S A153734 1,1,1,1,1,2,1,1,2,3,3,1,1,4,5,5,6,4,1,1,9,5,5,5,10,16,9,10,5,1,1,6,
%T A153734 14,14,35,15,21,21,14,20,35,14,15,6,1,1,7,20,14,21,28,56,64,70,42,14,
%U A153734 90,35,70,56,28,35,64,20,21,7,1
%N A153734 Triangle T(n,k): T(n,k) gives the A153452(m_k) such that A056239(m_k) = n, [1<=k<=A000041(n)], sorted by m_k, read by rows. Sequence A060240 is this sequence's permutation.
%C A153734 Lengths of rows are 1, 1, 2, 3, 5, 7, 11, 15, 22, 30,.... (A000041). Row sums give A000085.
%H A153734 Alois P. Heinz, <a href="/A153734/b153734.txt">Rows n = 0..26, flattened</a>
%e A153734 For n=4, A056239(7) = A056239(9) = A056239(10) = A056239(12) = A056239(16) = 4. Hence T(4,k) = A153452(m_k) = (1,2,3,3,1), where 1<=k<=5, m_k = 7,9,10,12,16.
%e A153734 Triangle T(n,k) begins:
%e A153734   1;
%e A153734   1;
%e A153734   1, 1;
%e A153734   1, 2, 1;
%e A153734   1, 2, 3, 3, 1;
%e A153734   1, 4, 5, 5, 6,  4,  1;
%e A153734   1, 9, 5, 5, 5, 10, 16, 9, 10, 5, 1;
%e A153734   ...
%p A153734 with(numtheory):
%p A153734 g:= proc(n) option remember; `if`(n=1, 1,
%p A153734       add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
%p A153734     end:
%p A153734 b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
%p A153734        [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
%p A153734     end:
%p A153734 T:= n-> map(g, sort(b(n, n)))[]:
%p A153734 seq(T(n), n=0..10);  # _Alois P. Heinz_, Aug 09 2012
%t A153734 g[n_] := g[n] = If[n == 1, 1, Sum[g[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]];
%t A153734 b[n_, i_] := b[n, i] = If[n == 0 || i < 2, {2^n}, Flatten[Table[Map[ #*Prime[i]^j&, b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A153734 T[n_] := g /@ Sort[b[n, n]];
%t A153734 T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, Feb 16 2021, after _Alois P. Heinz_ *)
%Y A153734 Cf. A067924, A215366.
%K A153734 easy,nonn,look,tabf
%O A153734 0,6
%A A153734 _Naohiro Nomoto_, Dec 31 2008