A391474 - OEIS

A391474

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

9, 15, 21, 25, 27, 33, 35, 45, 105, 770, 5950, 31815, 32445, 33705, 442365

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OFFSET

1,1

COMMENTS

Sequence A251820 is a subsequence of this sequence.

For all numbers k in the sequence found so far SRS(k) has either 3 or 4 parts and thus only one single positive difference of two parts.

The next entry in this sequence is larger than 10^7.

For numbers k computed so far in this sequence only differences 0, 2, 6, 8, 9 and 10 occur in SRS(k) while differences 1, 3, 4, 5 and 7 appear to occur only together with differences larger than 10.

LINKS

Table of n, a(n) for n=1..15.

EXAMPLE

Numbers belonging to this sequence:

A251820(1) = 15 since SRS(15) = { 8, 8, 8 }.

21 since in SRS(21) = { 11, 5, 5, 11 } number 6 is the only positive difference of two parts.

770 since SRS(770) = { 579, 570, 579 } has 9 as the only positive difference.

A251820(2) = 5950 since SRS(5950) = { 4464, 4464, 4464 }.

Number 117 is not in this sequence. Though SRS(117) = { 59, 21, 22, 21, 59 } has 1 as its smallest positive difference, the other positive differences exceed 10.

Table of first occurrence of difference 0 <= d <= 10 of any 2 parts for numbers k <= 10^7:

difference: 0 1 2 3 4 5 6 7 8 9 10

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smallest k: 15 - 9 - - - 21 - 25 45 33

MATHEMATICA

(* Function partsSRS[ ] is defined in A377654 *)

diffs[ps_, half_] := Union[Flatten[Map[Abs[ps-RotateLeft[ps, #]]&, Range[half]]]]

allDiffsPartsQ[k_, d_] := Module[{ps=partsSRS[k], len}, len=Length[ps]; Length[ps]>2&&AllTrue[diffs[ps, (len+Boole[OddQ[len]])/2 ], #<=d&]]

a391474[b_, d_] := Select[Range[b], allDiffsPartsQ[#, d]&]

a391474[33705, 10] (* long computation time for 442365 *)

CROSSREFS

Cf. A237270, A237271, A237593, A251820, A377654, A391475.