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

0,6

Comments

Lengths of rows are 1, 1, 2, 3, 5, 7, 11, 15, 22, 30,.... (A000041). Row sums give A000085.

Links

Alois P. Heinz, Rows n = 0..26, flattened

Example

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;
  ...

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 *)

Crossrefs

Cf. A067924, A215366.

Sequence in context: A253240 A290472 A060240 * A285554 A128495 A328062

Adjacent sequences: A153731 A153732 A153733 * A153735 A153736 A153737

Keyword

easy,nonn,look,tabf

Author

Naohiro Nomoto, Dec 31 2008

Status

approved