A377654 -id:A377654 - cp's OEIS frontend

A265999 Numbers k such that in the symmetric representation of sigma(k) all parts are of the same size.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 52, 53, 54, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 71, 72, 73, 74, 76, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 96, 97, 100

Offset: 1

Omar E. Pol, Dec 19 2015

nonn

All powers of 2, all prime numbers and all even perfect numbers are members of this sequence.
For more information about the symmetric representation of sigma see A237270 and A237593.
Sequence A174973: the symmetric representation of sigma, SRS(A174973(n)) consisting of 1 part, and sequence A239929: SRS(A239929(n)) consisting of 2 parts, are proper subsequences. Sequence A251820: SRS(A251820(n)) consisting of 3 equal parts, contains the only other known members 15 and 5950 of this sequence. No number m with SRS(m) consisting of 4 or more equal parts is known. - Hartmut F. W. Hoft, Jan 11 2025

			9 is not in the sequence because the parts of the symmetric representation of sigma(9) = 13 are [5, 3, 5].
10 is in the sequence because the parts of the symmetric representation of sigma(10) = 18 are [9, 9].
SRS(15) = { 8, 8, 8 } and SRS(5950) = { 4464, 4464, 4464 }. - _Hartmut F. W. Hoft_, Jan 11 2025

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A000079, A000203, A000396, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A245092, A262626, A266000.

Cf. A174973, A239929, A251820, A377654.

(* Function partsSRS[ ] is defined in A377654 *)
a265999[n_] := Select[Range[n], Length[Union[partsSRS[#]]]==1&]
a265999[100] (* Hartmut F. W. Hoft, Jan 11 2025 *)

A266000 Numbers k such that the symmetric representation of sigma(k) has at least two parts of distinct size.

Original entry on oeis.org

9, 21, 25, 27, 33, 35, 39, 45, 49, 50, 51, 55, 57, 63, 65, 69, 70, 75, 77, 81, 85, 87, 91, 93, 95, 98, 99, 105, 110, 111, 115, 117, 119, 121, 123, 125, 129, 130, 133, 135, 141, 143, 145, 147, 153, 154, 155, 159, 161, 165, 169, 170, 171, 175, 177, 182, 183, 185, 187, 189, 190, 195

Offset: 1

Omar E. Pol, Dec 19 2015

nonn

In other words: numbers k such that the symmetric representation of sigma(k) has at least two parts with distinct number of cells.
For more information about the symmetric representation of sigma see A237270 and A237593.
When the symmetric representation of sigma of m, SRS(m), consists of 2n-1 or 2n parts, n>=1, then at most n parts can be of distinct sizes. For the published terms in A239663, SRS(A239663(n)) consists of n parts representing ceiling(n/2) parts of distinct sizes, n>=1. Only two numbers m are known, 15 and 5950 in A251820, for which SRS(m) consists of n parts of less than ceiling(n/2) distinct sizes. - Hartmut F. W. Hoft, Jan 11 2025

			The symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts: 5 + 3 + 5 = 13, so 9 is in the sequence because the structure contains at least two parts of distinct size.
From _Hartmut F. W. Hoft_, Jan 11 2025: (Start)
SRS(a(1)) = SRS(A239663(3)) = SRS(9) = { 5, 3, 5 } is the smallest with 2 parts of distinct sizes.
SRS(a(14)) = SRS(A239663(5)) = SRS(63) = { 32, 12, 16, 12, 32 } is the smallest with 3 parts of distinct sizes.
SRS(a(127)) = SRS(A239663(7)) = SRS(357) = { 179, 61, 29, 38, 29, 61, 179 } is the smallest with 4 parts of distinct sizes. (End)

Complement of A265999.
Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A245092, A262626.
Cf. A239663, A251820, A377654.

(* Function partsSRS[ ] is defined in A377654 *)
a266000[n_] := Select[Range[n], Length[Union[partsSRS[#]]]>=2&]
a266000[200] (* Hartmut F. W. Hoft, Jan 11 2025 *)

Extended from a(37) to a(62) by Hartmut F. W. Hoft, Jan 11 2025

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

Original entry on oeis.org

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

Hartmut F. W. Hoft, Nov 27 2024

nonn

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.

			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.

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

(* 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]

A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 2, 3, 0, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 2, 1, 1, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 4, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0

Offset: 1

Omar E. Pol, Jun 29 2025

This sequence shares infinitely many terms with A067742 from which first differs at a(18). It also shares with A067742 the positions of zeros and nonzeros.
Observation: consider the 2-dense sublists of divisors of n. At least for the first 88 terms a(n) coincides with the number of odd terms in the central 2-dense sublist of divisors of n. For more information see A384225 and A280940.
See the "Discussion" text file in the first link for more comments.

			See the "Discussion" text file in the first link for the examples.

Omar E. Pol, Discussion of the Sequence A384230.
Omar E. Pol, Illustration of initial terms of A000203 in the pyramid.
Omar E. Pol, Illustration of initial terms of A001065 in the pyramid.
Omar E. Pol, Illustration of initial terms of A048050 in the pyramid.
Omar E. Pol, Illustration of initial terms of A067742 in the pyramid.
Omar E. Pol, Illustration of initial terms of A224613 (black spiders).
Omar E. Pol, Illustration of initial terms of A237593 (essentially a template for the Pop-Up pyramid).
Omar E. Pol, Prism of partitions of 10 and its companion tower (both have the same volume).
Omar E. Pol, The symmetric representation of sigma(n), n = 1..64 (the top view of the pyramid and of the tower).

Cf. A001227 (number of subparts), A071561 (positions of zeros), A071562 (positions of nonzeros), A237270 (parts), A237271, A237593, A279387 (subparts), A280940, A384225, A335574, A338488, A377654.

See the "Discussion" text file in the first link for more cross-references.

a(n) = 0 if and only if A067742(n) = 0.
a(n) >= A067742(n).
(a(n) - A067742(n)) is an even number.

Edited by Omar E. Pol, Aug 24 2025

A384704 Triangle T(i, j), 1 <= j <= i, read by rows. T(i, j) is the smallest number k that has i odd divisors and whose symmetric representation of sigma, SRS(k), has j parts; when no such k exists then T(i, j) = -1.

Original entry on oeis.org

1, 6, 3, 18, -1, 9, 30, 78, 15, 21, 162, -1, -1, -1, 81, 90, 666, 45, 75, 63, 147, 1458, -1, -1, -1, -1, -1, 729, 210, 1830, 135, 105, 165, 189, 357, 903, 450, -1, 225, -1, 1225, -1, 441, -1, 3025, 810, 53622, 405, -1, 1377, 1875, 567, 1539, 4779, 6875, 118098, -1, -1, -1, -1, -1, -1, -1, -1, -1, 59049

Offset: 1

Hartmut F. W. Hoft, Jun 07 2025

T(i, j) = -1 for i  >= 1 odd, nonprime, j even with 1 < j < i; also for i prime and all j with 1 < j < i.
The single value T(10, 4) = -1 has been verified; see the conjecture below.
T(i, i) <= 3^(i-1) for all i >=1 . Equality holds for all primes i. T(i, i) = A318843(i), for all i >= 1.
A038547(i) is the smallest number with exactly i odd divisors. Thus odd number A038547(i) occurs in row i of triangle T(i, j) so that A038547 is a subsequence of this sequence. For i prime, A038547(i) = T(i, i). For 4 <= i <= 10^9 nonprime, A038547(i) is in the third column, T(i, 3), except for i=8; furthermore, the first part of SRS(A038547(i)) has width 1 and size (A038547(i)+1)/2.
T(i, 1) <= 2 * 3^(i-1) and it is even for all i >1. Equality holds for all primes i.
T(i, 2) <= 2 * 3^(i/2-1) * p for all even i where p is the smallest prime greater than 4 * 3^(i/2-1). Equality holds when i = 2 * h where h is prime.
The positive numbers in columns 1..6 are subsequences of A174973, A239929, A279102, A280107, A320066, A320511, respectively.
Conjectures:
All entries T(i, j) in columns j >= 3 are odd.
T(i, 1)/2 is odd for all i > 1.
T(i, 1) = 2 * T(i, 3) for all nonprime i > 3, for i = 3, but not for i = 8.
T(i, 2)/2 is odd for all even i > 2.
T(i, 3) = A038547(i) for all nonprime i > 3, except i = 8.
T(2*i, 2*j) = -1 for j >= 2 and all prime i satisfying i >= prime(j+1).
From Omar E. Pol, Jun 08 2025: (Start)
T(i,j) is also the smallest number k whose symmetric representation of sigma(k) has i subparts and j parts, or -1 if no such k exists.
Observations:
At least for i < 12 if i is prime then T(i,1) = 2*T(i,i).
At least for i < 12 if i is prime then all terms in row i are -1's except the first and the last term. (End)

			The first 12 rows of triangle T(i, j):
   i\j      1     2   3   4    5    6    7    8    9   10    11    12
   1:       1
   2:       6     3
   3:      18    -1   9
   4:      30    78  15  21
   5:     162    -1  -1  -1   81
   6:      90   666  45  75   63  147
   7:    1458    -1  -1  -1   -1   -1  729
   8:     210  1830 135 105  165  189  357  903
   9:     450    -1  25  -1 1225   -1  441   -1 3025
  10:     810 53622 405  -1 1377 1875  567 1539 4779 6875
  11:  118098    -1  -1  -1   -1   -1   -1   -1   -1   -1 59049
  12:     630 16290 315 495  525 1071 1287 1197 2499 6069 13915 29095
  ...

Cf. A003056, A038547, A174973, A235791, A237048, A237591, A237593, A239929, A249223, A279102, A279387, A280107, A318843, A320066, A320511, A377654.

(* function partsSRS[ ] is defined in A377654 *)
setupT[d_] := Module[{list=Table[0, {i, d}, {j, i}], s, t}, For[s=1, s<=d, s++, For[t=1, t<=s, t++, If[(OddQ[s]&&Not[PrimeQ[s]]&&EvenQ[t]&&1

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A265999 Numbers k such that in the symmetric representation of sigma(k) all parts are of the same size.

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A266000 Numbers k such that the symmetric representation of sigma(k) has at least two parts of distinct size.

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

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A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

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A384704 Triangle T(i, j), 1 <= j <= i, read by rows. T(i, j) is the smallest number k that has i odd divisors and whose symmetric representation of sigma, SRS(k), has j parts; when no such k exists then T(i, j) = -1.

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