A238524 - OEIS

%I #33 Dec 13 2024 09:40:21

%S 3,5,7,9,10,11,13,14,15,17,19,21,22,23,25,26,27,29,31,33,34,35,37,38,

%T 39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,65,67,68,69,

%U 70,71,73,74,75,76,77,78,79,81,82,83,85,86,87,89,91,92

%N Numbers n such that the symmetric representation of sigma(n) is formed by two or more parts.

%C Complement of A174973.

%C First differs from A237046 at a(48).

%C First differs from A237287 at a(55).

%C For more information see A237270.

%C From _Hartmut F. W. Hoft_, Nov 27 2014: (Start)

%C Suppose n = 2^m * p1^e1 *...* pk^ek where p1 < ... < pk are the odd prime factors of n, m>=0 and all ej>0. Equivalent to the property of numbers in this sequence are:

%C (a) The number of 1's in odd positions equals the number of 1's in even positions in the n-th row of triangle A237048 through an index of the form 2^(m+1)*q where q is an odd divisor of n.

%C (b) There is one odd prime factor pj of n satisfying  pj > 2^(m+1) * product_{i<j}(pi^ei).

%C Also numbers n for which the n-th row in irregular triangle A249223 contains a zero.

%C (End)

%e 9 is in the sequence because the symmetric representation of sigma(9) = 13 is formed by three parts: [5, 3, 5], as shown below in the first quadrant:

%e .        5

%e .    _ _ _ _ _

%e .   |_ _ _ _ _|

%e .             |_ _ 3

%e .             |_  |       Sigma(9) = 5 + 3 + 5 = 13

%e .               |_|_ _

%e .                   | |

%e .                   | |

%e .                   | | 5

%e .                   | |

%e .                   |_|

%e .

%e From _Hartmut F. W. Hoft_, Nov 27 2014: (Start)

%e Number 78 = 2 * 3 * 13 has 1's in the 78th row of triangle A237048 at indices 1, 3, 4, 12 where 12 = 2^2*3 < 13. The symmetric representation of sigma(78) has two regions that meet at a point on the diagonal (width 0) and their third leg has width 2. Note also that 78 is the smallest number in this sequence for which width 0 occurs at an index that is not a power of 2.

%e (End)

%t (* sequence of numbers k for m <= k <= n having two or more parts *)

%t (* Function a237270[] is defined in A237270 *)

%t a238524[m_, n_]:=Select[Range[m, n], Length[a237270[#]]>=2&]

%t a238524[1, 260] (* data *)

%t (* _Hartmut F. W. Hoft_, Jul 07 2014 *)

%t (* function for the alternate description of the sequence *)

%t (* functions row[ ] & a237048[ ] are defined in A237048 *)

%t zero249223Q[n_] := Module[{i=2, bound=row[n], width=1}, While[width>=1 && i<=bound, width += (-1)^(i+1) * a237048[n, i]; i++]; width==0]

%t Select[Range[1, 100], zero249223Q] (* data *)

%t (* _Hartmut F. W. Hoft_, Nov 27 2014 *)

%Y Cf. A174973, A196020, A236104, A235791, A237046, A237287, A237591, A237593, A237270, A237271.

%K nonn

%O 1,1

%A _Omar E. Pol_, Mar 06 2014