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.
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, 14, 14, 35, 15, 21, 21, 14, 20, 35, 14, 15, 6, 1, 1, 7, 20, 14, 21, 28, 56, 64, 70, 42, 14, 90, 35, 70, 56, 28, 35, 64, 20, 21, 7, 1
Offset: 0
Examples
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. Triangle T(n,k) begins: 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; ...
Links
- Alois P. Heinz, Rows n = 0..26, flattened
Programs
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Maple
with(numtheory): g:= proc(n) option remember; `if`(n=1, 1, add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n))) end: b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n], [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)]) end: T:= n-> map(g, sort(b(n, n)))[]: seq(T(n), n=0..10); # Alois P. Heinz, Aug 09 2012 -
Mathematica
g[n_] := g[n] = If[n == 1, 1, Sum[g[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]]; 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[n_] := g /@ Sort[b[n, n]]; T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Feb 16 2021, after Alois P. Heinz *)
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