A391475 Numbers k for which the symmetric representation of sigma, SRS(k), has at least 3 parts and the smallest positive difference between two parts is at most 10, or all parts have the same size.

9, 15, 21, 25, 27, 33, 35, 45, 63, 81, 99, 105, 117, 153, 189, 325, 357, 399, 425, 475, 483, 609, 651, 715, 770, 777, 783, 825, 837, 861, 897, 999, 1107, 1161, 1221, 1269, 1353, 1419, 1431, 1539, 1551, 1595, 1705, 1749, 1833, 1863, 1947, 2013, 2035, 2067, 2301, 2379, 2405, 2541, 2613, 2665, 2769, 2795

Offset

1,1

Comments

For any k in this sequence for which SRS(k) is either { a, b, a } or { a, b, b, a }, abs(b-a) is the only possible positive difference when a != b. When SRS(k) has 5 or more parts several differences are possible.

This sequence includes the two known numbers 15 and 5950 in sequence A251820.

Two positive differences d_1 < d_2 <= 12 do not exist for SRS(k) with k <= 10^6.

Links

Table of n, a(n) for n=1..58.

Example

Numbers belonging to the sequence:
A251820(1) = 15 since SRS(15) = { 8, 8, 8 }.
63 since SRS(63) = { 32, 16, 12, 16, 32 } has 4 as least positive difference of two parts.
837 since SRS(837) = { 419, 141, 51, 58, 51, 141, 419 } has 7 as least positive difference of two parts.
A251820(2) = 5950 since SRS(5950) = { 4464, 4464, 4464 }.
6325 since SRS(6325) = { 3163, 635, 293, 288, 85, 85, 288, 293, 635, 3163 } has 5 as least positive difference of two parts.
Table of first occurrence of smallest positive difference:
  difference:  0    1    2    3    4    5    6    7    8    9    10
  -----------------------------------------------------------------
  smallest k:  15   117  9    325  63   6325 21   837  25   45   33
Numbers 9, 15, 21, 25, 33 and 45 are members of A391474.

Mathematica

(* Function partsSRS[ ] is defined in A377654 *)
diffs[ps_, half_] := Union[Flatten[Map[Abs[ps - RotateLeft[ps, #]] &, Range[half]]]]
leastDiffQ[k_, d_] := Module[{ps = partsSRS[k], len, lDiff}, len = Length[ps]; lDiff = diffs[ps, (len + Boole[OddQ[len]])/2]; Length[ps] > 2 && (Intersection[Range[d], lDiff] != {} || lDiff == {0})]
a391475[b_] := Select[Range[b], leastDiffQ[#, 10] &]
a391475[2795]

Crossrefs

Cf. A237270, A237271, A237593, A251820, A377654, A391474.

Sequence in context: A046446 A390760 A046350 * A391474 A225438 A175076

Adjacent sequences: A391472 A391473 A391474 * A391476 A391477 A391478

Keyword

nonn

Author

_Hartmut F. W. Hoft_, Dec 10 2025

Status

approved