A390627 - OEIS

A390627

Numbers k such that the symmetric representation of sigma, SRS(k), and SRS(2k), both considered as lists of the sizes of their parts, have the same number of parts and SRS(2k) is an integer parts-multiple of SRS(k).

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203

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OFFSET

1,2

COMMENTS

By definition, this sequence is a subsequence of A390626.

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

The sum of the parts of SRS(k) equals sigma(k) - see the link in A280851 for a proof. Therefore, if SRS(2k) is an integer parts-multiple of SRS(k) for a k in this sequence then k is odd, because for k = 2^m * q with m >= 0 and q odd, sigma(2k) / sigma(k) = (2^(m+2) - 1) / (2^(m+1) - 1) equals 3 when m = 0.

Suppose that odd number k has 3 as a divisor. Then the first part of SRS(2k) is greater than 3*(k+1)/2, the length of its first 3 legs, since the 3rd leg has width 2, while the first part of SRS(k) has size (k+1)/2. This contradicts the integral parts-multiplier of 3 for k to be a number in this sequence.

Note that this sequence differs from A390628 first with a(377) = 1309.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

Number 5 belongs to the sequence since SRS(10) = { 9, 9 } and SRS(5) = { 3, 3 }; both have width pattern 1 0 1.

Number 25 is the smallest number for which SRS(25) = { 13, 5, 13 } consists of 3 parts. SRS(50) = { 39, 15, 39 } has the integer parts-multiplier of 3 and both 25 and 50 have width pattern 1 0 1 0 1.

Therefore, numbers 5 and 25 also belong to sequence A390628.

Number 1309 = 7 * 11 * 17 is the smallest number k in this sequence for which SRS(2k) and SRS(k) have different width patterns:

SRS(2618) = { 1965, 627, 627, 1965 }; width pattern of 2618: 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 parts-unimodal

SRS(1309) = { 655, 209, 209, 655 }; width pattern of 1309: 1 0 1 2 1 2 1 0 1 2 1 2 1 0 1 not parts-unimodal

Number 2275 = 5^2 * 7 * 13 is the smallest number k in this sequence for which neither SRS(k) nor SRS(2k) is not parts-unimodal:

SRS(4450): 1 0 1 2 3 2 3 2 3 2 3 4 3 2 3 2 3 2 3 2 1 0 1

SRS(2275): 1 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 0 1

Therefore, numbers 1309 and 2275 do not belong to sequence A390628.

MATHEMATICA

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

partsQ[k_] := Module[{pS2=partsSRS[2k], pS1=partsSRS[k]}, Length[pS2]==Length[pS1]&& Length[Union[pS2/pS1]]==1&&AllTrue[pS2/pS1, IntegerQ]]

a390627[b_] := Select[Range[b], partsQ]

a390627[203]

CROSSREFS

Cf. A237270, A237271, A237593, A280851, A377654, A390626, A390628.

Sequence in context: A007310 A069040 A070191 * A389633 A367734 A231810

Adjacent sequences: A390624 A390625 A390626 * A390628 A390629 A390630

KEYWORD

nonn

AUTHOR

Hartmut F. W. Hoft, Nov 13 2025

STATUS

approved