A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105

Offset: 1

Author

Hartmut F. W. Hoft, Nov 27 2024

Keywords

• nonn

Comments

Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).

This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.

A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.

Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.

For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.

Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.

Examples

			a(5) = 10 is in the sequence since SRS(10) = {9, 9} consists of 2 parts of width 1 and of sizes 9 = (2^2 - 1)(5+1)/2.
a(15) = 25 is in the sequence since the first part of SRS(25) = {13, 5, 13} has width 1 and has size 13 = (2^1 - 1)(25+1)/2.
a(28) = 44 is in the sequence since SRS(44) = {42, 42} has width 1 and has size 42 = (2^3 - 1)(11+1)/2.
The upper left hand 11 X 11 section of array T(j, k) shows the j-th number m in this sequence of the form m = 2^k * q with q odd. The first part of SRS(m) of every number in column k consists of 2^(k+1) - 1 legs of width 1.
j\k| 0   1   2    3    4     5     6      7      8       9       10  ...
------------------------------------------------------------------------
1  | 3   10  44   136  592   2144  8384   32896  133376  527872  2102272
2  | 5   14  52   152  656   2272  8768   33664  133888  528896  2112512
3  | 7   22  68   184  688   2336  8896   34432  138496  531968  2118656
4  | 9   26  76   232  752   2528  9536   34688  140032  537088  2130944
5  | 11  34  92   248  848   2656  9664   35456  142592  538112  2132992
6  | 13  38  116  296  944   2848  10048  35968  144128  543232  2137088
7  | 15  46  124  328  976   3104  10432  36224  145664  544256  2139136
8  | 17  50  148  344  1072  3232  10688  37504  146176  547328  2149376
9  | 19  58  164  376  1136  3296  11072  39296  147712  556544  2161664
10 | 21  62  172  424  1168  3424  11456  39808  150272  558592  2163712
11 | 23  70  188  472  1264  3488  11584  40064  151808  559616  2180096
...
Row 1 is A246956(n), n>=1.
Column 0 is A005408(n) with T(j + 1, 0) = T(j, 0) + 2, n>=1.
Column 1 is A091999(n) with T(j + 2, 1) = T(j, 1) + 12, n>=2.
Column 2 is A270298(n) with T(j + 48, 2) = T(j, 2) + 840, n>=1.
Column 3 is A270301(n) with T(j + 5760, 3) = T(j, 3) + 240240, n>=1.
		

Crossrefs

Cf. A000079, A005408, A091999, A235791, A237287, A237591, A237593, A238524, A246956, A270298, A270301, A341969, A370206, A377654.

Programs

Mathematica

(* partsSRS[] and widthPattern[ ] are defined in A377654 *)
a378471[m_, n_] := Select[Range[m, n], Length[partsSRS[#]]>1&&widthPattern[#][[1;;2]]=={1, 0}&]
a378471[1, 105]