

A238524


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


10



3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92
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OFFSET

1,1


COMMENTS

First differs from A237046 at a(48).
First differs from A237287 at a(55).
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:
(a) The number of 1's in odd positions equals the number of 1's in even positions in the nth row of triangle A237048 through an index of the form 2^(m+1)*q where q is an odd divisor of n.
(b) There is one odd prime factor pj of n satisfying pj > 2^(m+1) * product_{i<j}(pi^ei).
Also numbers n for which the nth row in irregular triangle A249223 contains a zero.
(End)


LINKS



EXAMPLE

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:
. 5
. _ _ _ _ _
. _ _ _ _ _
. _ _ 3
. _  Sigma(9) = 5 + 3 + 5 = 13
. __ _
.  
.  
.   5
.  
. _
.
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.
(End)


MATHEMATICA

(* sequence of numbers k for m <= k <= n having two or more parts *)
(* Function a237270[] is defined in A237270 *)
a238524[m_, n_]:=Select[Range[m, n], Length[a237270[#]]>=2&]
a238524[1, 260] (* data *)
(* function for the alternate description of the sequence *)
(* functions row[ ] & a237048[ ] are defined in A237048 *)
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]
Select[Range[1, 100], zero249223Q] (* data *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



