A182743 Table of the partitions that do not contain 1 as a part for odd integers.

3, 5, 2, 4, 2, 2, 7, 3, 2, 2, 3, 2, 2, 2, 2, 6, 3, 2, 2, 2, 2, 5, 3, 3, 2, 2, 2, 2, 9, 4, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 8, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,1

Comments

The same idea as A182742 but for odd integers.

Links

Table of n, a(n) for n=1..91.

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Example

Array begins:
  3,2,2,2,2,2,2,2,2,2,2,
  5,2,2,2,2,2,2,2,2,2,
  4,3,2,2,2,2,2,2,2,2,
  7,2,2,2,2,2,2,2,2,
  3,3,3,2,2,2,2,2,2,2,
  6,3,2,2,2,2,2,2,2,
  5,4,2,2,2,2,2,2,2,
  9,2,2,2,2,2,2,2,
  5,3,3,2,2,2,2,2,2,
  4,4,3,2,2,2,2,2,2,
  8,3,2,2,2,2,2,2,
  7,4,2,2,2,2,2,2,
  6,5,2,2,2,2,2,2,
  11,2,2,2,2,2,2,
  4,3,3,3,2,2,2,2,2,
  7,3,3,2,2,2,2,2,
  6,4,3,2,2,2,2,2,
  5,5,3,2,2,2,2,2,
  10,3,2,2,2,2,2,
  5,4,4,2,2,2,2,2,
  9,4,2,2,2,2,2,
  8,5,2,2,2,2,2,
  7,6,2,2,2,2,2,

Maple

cmpL := proc(a, b) local i ; for i from 1 to min(nops(a), nops(b)) do if op(i, a) < op(i, b) then return -1 ; elif op(i, a) > op(i, b) then return 1 ; end if; end do; if nops(a) > nops(b) then return 1; elif nops(a) < nops(b) then return -1; else return 0; end if; end proc:
pShellMin := proc(p) local idx, j; idx := 1 ; for j from 2 to nops(p) do if cmpL( op(j, p), op(idx, p)) < 0 then idx := j; end if; end do; return idx ; end proc:
A141285rowf := proc(n) local p; if n <= 1 then [n] ; else psort := [] ; p := combinat[partition](n) ; while nops(p) > 0 do m := pShellMin(p) ; mmi := min(op(op(m, p))) ; if mmi > 1 then mma := max(op(op(m, p))) ; psort := [op(psort), sort(op(m, p), `>`)] ; end if; p := subsop(m=NULL, p) ; end do: psort ; end if; end proc:
for n from 1 to 17 by 2 do shl := A141285rowf(n) ; for r in shl do for k in r do printf("%d, ", k) ; end do: printf("\n") ; end do: printf("\n") ; end do: # R. J. Mathar, Dec 03 2010

Prog

(PARI)
a(max_part)={ /* e.g. a(12) for A182742, a(13) for A182743 */
  my(p = vecsort(partitions(max_part, [2, max_part])));
  for(i = 1, #p,
    for(j=1, max(#p[i]+1, min(#p+1-i, 20)),
      my(k=#p[i]+1-j);
      print1(if(k>0, p[i][k], 2), ", "));
    print("...");
  );
  print("...");
  concat(vector(#p, i, vector(i, j, my(k=#p[i+1-j]+1-j); if(k>0, p[i+1-j][k], 2))))
}; /* Martin Fuller, Jan 15 2026 */

Crossrefs

Cf. A135010, A138121, A141285, A182731, A182742.

Column 1 give A182733. Column 2 give A182745.

Sequence in context: A261597 A197331 A113475 * A222601 A104807 A309492

Adjacent sequences: A182740 A182741 A182742 * A182744 A182745 A182746

Keyword

nonn,tabl

Author

Omar E. Pol, Nov 30 2010, Dec 01 2010

Extensions

a(56) onward from Martin Fuller, Jan 15 2026

Status

approved