A394935 a(n) is the smallest k for which the number of parts of the symmetric representation of sigma of k, SRS(k), is odd and the lower Dyck path of SRS(k) touches the diagonal at the same point as A298856(k).

4, 12, 24, 56, 80, 108, 140, 176, 255, 352, 408, 468, 598, 667, 744, 825, 910, 1085, 1276, 1380, 1715, 1833, 1960, 2214, 2491, 2632, 2784, 2928, 3082, 3404, 3744, 3920, 4466, 4661, 5056, 5254, 5673, 5888, 6106, 6336, 7029, 7504, 7752, 8004, 8520, 9047, 9317, 9592

Offset

1,1

Comments

k=2 is the smallest index for which a(k) > A298856(k) + 1.

Links

Table of n, a(n) for n=1..48.

Example

For n = 1, 2: A394934(1) = 2 and A394934(2) = 9 below A298856(1) = 3 and A298856(2) = 10 respectively, and a(1) = 4 and a(2) = 12 above the latter two.
  _ _ _ _ _ _ _
12|_ _ _ _ _ _  |
  ._ _ _ _ _ _| |_ _
10|_ _ _ _ _  |     |_
  |_ _ _ _ _| |_      |
  .         |_ _|_    |_ _
  .         |_  | |_ _ _  |
  .           |_|_ _  | | |
  ._ _ _          | | | | |
  |_ _  |_        | | | | |
 3|_ _|_  |       | | | | |
  |_  | | |       | | | | |
  . |_|_|_|. . . .|_|_|.|_|
     2 3 4        9 10  12

Mathematica

(* support functions restate functions in the respective sequences *)
row[n_] := Floor[(Sqrt[8 n+1]-1)/2]
a014105[n_] := n(2n+1)
a240542[n_] := Sum[(-1)^(k+1) Ceiling[(n+1)/k-(k+1)/2], {k, 1, row[n]}]
t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, row[n]]]
a298856[b_] := Select[Map[a014105, Range[b]], a240542[#]==a240542[#-1]&] (* b bound of search *)
nextOddSRS[n_] := NestWhile[#+1&, n+1, t249223[#][[row[#]]]==0&]
a394935[b_] := Map[nextOddSRS, a298856[b]]
a394935[70]

Crossrefs

Cf. A014105, A071562, A237593, A240542, A249223, A298856, A394934.

Sequence in context: A136486 A003203 A051193 * A356550 A216244 A215223

Adjacent sequences: A394932 A394933 A394934 * A394936 A394938 A394939

Keyword

nonn

Author

Hartmut F. W. Hoft, Apr 07 2026

Status

approved