A390628 Numbers k such that the symmetric representation of sigma, SRS(k), and SRS(2k) have the same width pattern.

1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 46, 47, 48, 49, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 68, 71, 72, 73, 74, 76, 77, 79, 80, 82, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 100

Offset

1,2

Comments

This sequence is a subsequence of A390626 since the number of contiguous sections of positive numbers in the width pattern of a number k equals the number of parts of SRS(k).

The terms "parts-multiple", "integer parts-multiple" and "parts-unimodal" are defined in A390626.

Number 30 is the smallest in A390626 that is not in this sequence.

Links

Table of n, a(n) for n=1..66.

Example

Number 6 is in this sequence since 6 and 12 have the width pattern 1 2 1;
SRS(6) = { 12 } and SRS(12) = { 28 }; number 6 is not in A390627.
Number k = 18 is the smallest k in this sequence for which SRS(18) = { 39 } is not (parts)-unimodal: 1 2 1 2 1.
Number 174 is the smallest k in this sequence for which SRS(k) has 2 parts and maximum width 2; SRS(348) = { 420, 420 } and SRS(174) = { 180, 180 } and their common width pattern is 1 2 1 0 1 2 1 ; number 174 is not in A390627.
Number 35 is the smallest k in this sequence for which SRS(k) has 3 parts and maximum width 2, and since 35 and 70 have the width pattern 1 0 1 2 1 0 1;
SRS(35) = { 18, 12, 18 } and SRS(70) = { 54, 36, 54 } so that number 35 is in A390627.
Number 1001 is the smallest k in this sequence for which SRS(k) has 4 parts and maximum width 3; numbers 2002 and 1001 have the width pattern 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1;
SRS(1001) = { 501, 171, 171, 501 } and SRS(2002) = { 1503, 513, 513, 1503 } so that number 1001 is in A390627.
Number 1314 is the smallest k in this sequence for which SRS(k) = { 3367, 3367 } has 2 parts that are not parts-unimodal: 1 2 1 2 1 0 1 2 1 2 1.
Number 7502 is the smallest k in this sequence for which SRS(7502) = { 5628, 1512, 5628 } has 3 parts with the central part not parts-unimodal: 1 0 1 2 1 2 1 2 1 0 1.

Mathematica

(* Function widthPattern[ ] is defined in A377654 *)
a390628[b_]:=Select[Range[b], widthPattern[#]==widthPattern[2#]&]
a390628[100]

Crossrefs

Cf. A237270, A237271, A237593, A377654, A390626, A390627.

Sequence in context: A194831 A111688 A390626 * A058049 A091871 A303393

Adjacent sequences: A390625 A390626 A390627 * A390629 A390630 A390631

Keyword

nonn

Author

Hartmut F. W. Hoft, Nov 13 2025

Status

approved