A376829 -id:A376829 - cp's OEIS frontend

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

1, 9, 25, 49, 81, 121, 169, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1369, 1521, 1681, 1849, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 4225, 4489, 4761, 5041, 5329, 6241, 6561, 6889, 7225, 7569, 7921, 8649, 9025, 9409, 10201, 10609, 11449, 11881, 12321, 12769, 13225, 14161, 14641, 15129, 15625

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

nonn

Since for numbers m^2 in the sequence the width at the diagonal of SRS(m^2) is 1, the area m of its center part is odd so that this sequence is a proper subsequence of A016754 and since SRS(m^2) has an odd number of parts it is a proper subsequence of A319529. The smallest odd square not in this sequence is 225 = 15^2.  SRS(225) is {113, 177, 113}, its center part has maximum width 2, its width at the diagonal is 1.
The k+1 parts of SRS(p^(2k)), p an odd prime and k >= 0, through the diagonal including the center part have areas (p^(2k-i) + p^i)/2 for 0 <= i <= k. They form a strictly decreasing sequence. Since p^(2k) has 2k+1 divisors and SRS(p^(2k)) has 2k+1 parts, all of width 1 (A357581), the even powers of odd primes form a proper subsequence of A244579. For the subsequence of squares of odd primes p, SRS(p^2) consists of the 3 parts { (p^2 + 1)/2, p, (p^2 + 1)/2 } see A001248, A247687 and A357581.
The areas of the parts of SRS(m^2) need not be in descending order through the diagonal as a(112) = 275^2 = 75625 with SRS(75625) = (37813, 7565, 3443, 1525, 715, 738, 275, 738, 715, 1525, 3443, 7565, 37813) demonstrates.
An equivalent description of the sequence is: The center part of SRS(m^2) has width 1, m is odd, and A249223(m^2, m-1) = 0.
Conjectures (true for all a(n) <= 10^8):
(1) The central part of SRS(a(n)) is the minimum of all parts of SRS(a(n)), 1 <= n.
(2) The terms in this sequence are the squares of the terms in A244579.

			The center part of SRS(a(3)) = SRS(25) has area 5, all 3 parts have width 1, and 25 with 3 divisors also belongs to A244579.
The center part of SRS(a(7)) = SRS(169) has area 13, all 3 parts have width 1, and 169 with 3 divisors also belongs to A244579.
The center part of SRS(a(10)) = SRS(441) has area 21 and width 1, but the maximum width of SRS(441) is 2. Number 441 has 9 divisors and SRS(441) has 7 parts while 21 has 4 divisors and SRS(21) has 4 parts so that 21 is in A244579 while 441 is not.

Cf. A001248, A030514, A030516, A030629, A030631, A179645 (even prime power sequences).

Cf. A003056, A016754, A237591, A237593, A244579, A247687, A249223, A319529, A341969, A357581.

(* t237591 and partsSRS compute rows in A237270 and A237591, respectively *)
(* t249223 and widthPattern are also defined in A376829 *)
row[n_] := Floor[(Sqrt[8 n+1]-1)/2]
t237591[n_] := Map[Ceiling[(n+1)/#-(#+1)/2]-Ceiling[(n+1)/(#+1)-(#+2)/2]&, Range[row[n]]]
partsSRS[n_] := Module[{widths=t249223[n], legs=t237591[n], parts, srs}, parts=widths legs; srs=Map[Apply[Plus, #]&, Select[SplitBy[Join[parts, Reverse[parts]], #!=0&], First[#]!=0&]]; srs[[Ceiling[Length[srs]/2]]]-=Last[widths]; srs]
t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, row[n]]]
widthPattern[n_] := Map[First, Split[Join[t249223[n], Reverse[t249223[n]]]]]
centerQ[n_] := Module[{pS=partsSRS[n]}, Sqrt[n]==pS[[(Length[pS]+1)/2]]]/;OddQ[n]
widthQ[n_] := Module[{wP=SplitBy[widthPattern[n], #!=0&]}, wP[[(Length[wP]+1)/2]]]=={1}/;OddQ[n]
a377654[m_, n_] := Select[Map[#^2&, Range[m, n, 2]], centerQ[#]&&widthQ[#]&]/;OddQ[m]
a377654[1, 125]

A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).

Original entry on oeis.org

60, 10728, 210, 315, 7620, 810, 495, 1155, 840456, 2070, 525, 28158, 945, 88410, 7290, 1275, 1995, 30555, 1575, 408150, 12810, 1287, 2625, 3003, 22365, 2835, 1313010, 45450, 6105, 3315, 10659, 18975, 382305, 11385

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

When m is odd and SRS(m) has maximum width 3 then SRS(m) has at least 3 parts because the first and last parts of SRS(m) consist of a single leg of width 1. Therefore, the first two rows of the table contain only even numbers. The numbers in the third row appear to be odd and divisible by 15.

			a(8) = T(3,2) = 1155 is the smallest example whose symmetric representation of sigma has 3 parts and 4 counts of width 3 in its width pattern.
Upper left hand section of table T(i, j) = m, numbers m <= 10^7, Columns j indicate 2j occurrences of width 3 in the width pattern of m. T(2, 7) > 10^7.
i\j| 1       2       3       4       5       6       7       8    ...
---------------------------------------------------------------------
1  | 60      210     810     2070    7290    12810   45450   146610
2  | 10728   7620    840456  88410   408150  1313010 >10^7   8596710
3  | 315     1155    945     1575    2835    11385   8505    40095
4  | 495     28158   30555   22365   382305  296835  256095  199395
5  | 525     1995    3003    18975   15147   23925   14553   186219
6  | 1275    2625    10659   35217   132957  818363  312039  1760031
7  | 1287    3315    13125   37107   44289   195415  482937  258687
8  | 6105    3861    31875   65625   132153  149435  807495  1426113
9  | 3591    10773   56889   66861   254065  797979  319599  2199477
10 | 6783    16443   57477   222999  417175  1540875 768339  4670991
11 | 18963   35397   106191  965979  1025973 1770783 2489151 7547427
12 | 90801   58653   47481   1223365 2449785 4600617 ...     ...
13 | 152019  107457  817209  2213253 1740081 4310481
14 | 257397  297087  410571  3086349 3552213 5170055
15 | 335225  815409  1360989 2079609 ...     ...
16 | 1523319 2600283 1642557 2563239
17 | 1473725 1739375 4116777 ...
18 | 4008125 3826625 3687475
19 | 7576085 7937875 ...
...  ...     ...

Subsequence of A376829.

Cf. A237591, A237593, A249223, A341969.

(* widthPattern[ ] and its support functions are defined in A376829 *)
t377667[b_, {r_, c_}] := Module[{t=ConstantArray[0, {r, c}], k, wP, c3, p3}, For[k=1, k<=b, k++, wP=widthPattern[k]; If[Max[wP]==3, c3=Count[wP, 3]; If[EvenQ[c3]&&c3<=2c, c3/=2; p3=Length[Select[SplitBy[wP, #!=0&], First[#]!=0&]]; If[p3<=r &&t[[p3, c3]]==0, t[[p3, c3]]=k]]]]; t]
t377667[1540875, {10, 6}] (* complete 10 x 6 upper left hand section of table *)

A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2i-1 parts and has 2j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

Original entry on oeis.org

72, 2450, 648, 1225, 120050, 450, 3969, 581042, 211250, 20808, 9801, 30625

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

Maximum width 3 can occur an odd number of times in the width pattern of SRS(m) only for numbers m in this sequence for which SRS(m) has an odd number of parts. In that case width 3 must occur at the diagonal of SRS(m). However, the center part of SRS(m) need not be unimodal.

			For a(1) = 72 SRS(a(1)) is unimodal: 12321.
For a(2) = 2450 the center part of SRS(a(2)) is not unimodal: 1212123212121.
For a(11) = 9801 SRS(a(11)) consists of 9 unimodal parts with maximum width in successive parts nondecreasing to the center part of SRS(a(11)); its width pattern is: 1 0 1 0 1 2 1 0 1 2 1 0 1 2 3 2 1 0 1 2 1  0 1 2 1 0 1 0 1.
Ragged upper left hand section of table T(i, j) = m, numbers m <= 10^7, rows i denoting 2*i-1 parts in SRS(m) and columns j denoting 2*j-1 occurrences of width 3 in the width pattern of SRS(m):
i\j  1       2       3       4       5       6       7    ...
-------------------------------------------------------------
1  | 72      648     450     20808   27378   11250   1996002
2  | 2450    120050  211250  61250   81225   5281250 1531250
3  | 1225    581042  >10^7   354025  >10^7   148225  442225
4  | 3969    30625   321489  127449  1500625 2393209
5  | 9801    6175225 765625  1375929         648025
6  | 4809249 88209   2082249 983961
7  | 385641  1185921 159201  >10^7
8  | 5461569 3470769         7144929
9  | 7177041
10 | 8497225
...

Subsequence of A376829.

Cf. A237591, A237593, A249223, A341969, A377667.

(* widthPattern[ ] and its support functions are defined in A376829 *)
t377668[b_, {r_, c_}] := Module[{t=ConstantArray[0, {r, c}], k, wP, c3, p3}, For[k=1, k<=b, k++, wP=widthPattern[k]; If[Max[wP]==3, c3=Count[wP, 3]; If[OddQ[c3]&&c3+1<=2c, c3=(c3+1)/2; p3=Length[Select[SplitBy[wP, #!=0&], First[#]!=0&]]; If[OddQ[p3]&&p3+1<=2r, p3=(p3+1)/2; If[t[[p3, c3]]==0, t[[p3, c3]]=k]]]]]; t]
t377668[581042, {4, 4}] (* initial 4x4 section except for T(3, 3) > 10^7 *)

cp's OEIS Frontend

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

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A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).

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A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2i-1 parts and has 2j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

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cp's OEIS Frontend

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

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Mathematica

A377667 Square array read by antidiagonals upwards: T(i,j) is the smallest number m such that the symmetric representation of sigma, SRS(m), has maximum width 3, consists of i parts and has 2*j occurrences of maximum width 3 in its width pattern (row m of A341969).

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A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2*i-1 parts and has 2*j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

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A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2i-1 parts and has 2j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).