A241559 -id:A241559 - cp's OEIS frontend

A241558 Smallest part of the symmetric representation of sigma(n).

1, 3, 2, 7, 3, 12, 4, 15, 3, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 5, 18, 12, 60, 5, 21, 6, 56, 15, 72, 16, 63, 7, 27, 12, 91, 19, 30, 8, 90, 21, 96, 22, 42, 23, 36, 24, 124, 7, 15, 10, 49, 27, 120, 8, 120, 11, 45, 30, 168, 31, 48, 12, 127, 9, 144, 34, 63, 13

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

If A237271(n) = 1 then a(n) = A241559(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241559(n) = A241838(n).
For more information see A237270 and A237593.

			For n = 9 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] and the smallest part is 3 so a(9) = 3.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the smallest part is 23 so a(45) = 23.
For n = 63 the symmetric representation of sigma(63) = 104 has five parts [32, 12, 16, 12, 32] and the smallest part is 12 so a(63) = 12.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241559, A241838, A245092, A262626.

(* Function a237270[] is defined in A237270 *)
a241558[n_]:=Min[a237270[n]]
Map[a241558,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

More terms from Jinyuan Wang, Feb 14 2020

A241838 Column 1 of A237270, also the right border.

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, 18, 12, 60, 13, 21, 14, 56, 15, 72, 16, 63, 17, 27, 18, 91, 19, 30, 20, 90, 21, 96, 22, 42, 23, 36, 24, 124, 25, 39, 26, 49, 27, 120, 28, 120, 29, 45, 30, 168, 31, 48, 32, 127

Offset: 1

Omar E. Pol, Apr 29 2014

First differs from A241559 at a(45).
If A237271(n) = 1 then a(n) = A241558(n) = A241559(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241558(n) = A241559(n).
For more information see A237593.

			For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23], both the first and the last term are equal to 23, so a(45) = 23.

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241558, A241559, A245092, A262626.

Map[First[a237270[#]]&,Range[64]] (* data : computing all parts *)
(* computing only the first part of the symmetric representation of sigma(n) *)
row[n_] := Floor[(Sqrt[8n+1]-1)/2] (* in A237591 *)
f[n_, k_] := If[Mod[n-k*(k+1)/2, k]==0, (-1)^(k+1), 0]
g[n_, k_] := Ceiling[(n+1)/k-(k+1)/2] - Ceiling[(n+1)/(k+1)-(k+2)/2] (* in A237591 *)
a241838[n_] := Module[{r=row[n], widths={}, i=1, w=0, len, legs}, w+=f[n, i]; While[i<=r && w!=0, AppendTo[widths, w]; i++; w+=f[n, i]]; len=Length[widths]; legs=Map[g[n, #]&, Range[len]]; If[lenHartmut F. W. Hoft, Jan 25 2018 *)

a(n) = A237270(n, 1) = A237270(n, A237271(n)).

A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145

Offset: 1

Omar E. Pol, Jul 02 2014

nonn

Numbers n such that A243982(n) = 0.
First differs from A151991 at a(25).
Let n = 2^m * q with m >= 0 and q odd. Let c_n denote the count of regions in the symmetric representation of sigma(n), which is determined by the positions of 1's in the n-th row of A237048. The maximum of c_n occurs when odd and even positions of 1's alternate implying that all regions have width 1, denoted by w_n = 1. When m > 0 then sigma_0(n) > sigma_0(q) and c_n = sigma_0(n) is impossible. Therefore, exactly those odd n with w_n = 1 are in this sequence. Furthermore, since the 1's in A237048 represent the odd divisors of n, their odd-even alternation expresses the property 2*f < g for any two adjacent divisors f < g of odd number n; in other words, this sequence is also the complement of A090196 relative to the odd numbers. This last property permits computations of elements in this sequence faster than with function a244579, which is based on Dyck paths. - Hartmut F. W. Hoft, Oct 11 2015
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
Also, integers n such that for any pair a < b of divisors of n the inequality 2*a < b holds (hence n is odd).
Let 1 = d_1 < ... < d_k = n be all (odd) divisors of n. The property 2*d_i < d_(i+1), for 1 <= i < k, is equivalent for the 1's in the n-th row of A249223 to be in positions 1 = d_1 < 2 < d_2 < 2*d_2 < ... < d_i <2*d_i < d_(i+1) < ... where 2*d_i represents the odd divisor e_i with d_i * e_i = n. In other words, the odd divisors are the number of parts in the symmetric representation of sigma(n). The rightmost 1 in the n-th row occurs in an odd (even) position when k is odd (even).
As a consequence this sequence is also the complement of A090196 in the set of odd numbers. (End)

			9 is in the sequence because the parts of the symmetric representation of sigma(9) are [5, 3, 5] and the divisors of 9 are [1, 3, 9] and in both cases there is the same number of elements: A237271(9) = A000005(9) = 3.
See the link for a diagram of the symmetric representations of sigma for sequence data listed above. The symmetric representations of sigma(a(35)) = sigma(81) = sigma(3^4) consists of 5 regions whose areas are [41, 15, 9, 15, 41] and computed as 41 = (3^4+3^0)/2, 15 = (3^3+3^1)/2, and 9 = 3^2 for the central area. Observe also that the 81st row in triangle A237048 is [ 1 1 1 0 0 1 0 0 1 0 0 0 ] with the 1's in positions 1, 2, 3, 6, and 9. This is the largest count for the symmetric regions of sigma shown in the diagram. - _Hartmut F. W. Hoft_, Oct 11 2015

Hartmut F. W. Hoft, Illustration of the symmetric representations of sigma for sequence data

Cf. A000005, A001227, A005279, A071561, A071562, A090196, A000203, A196020, A236104, A237048, A237270, A237271, A237593, A238443, A238524, A239657, A239929, A239663, A240062, A241558, A241559, A243982, A245092.

(* Function a237270[] is defined in A237270 *)
a244579[m_, n_] := Select[Range[m,n], Length[a237270[#]] == Length[Divisors[#]]&]
a244579[1, 150] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)
(* alternative function using the divisor property *)
divisorPairsQ[n_] := Module[{d=Divisors[n]}, Select[2*Most[d] - Rest[d], # >= 0&] == {}]
a244579Alt[m_?OddQ, n_] := Select[Range[m, n, 2], divisorPairsQ]
a244579Alt[1, 145] (* data *)
(* Hartmut F. W. Hoft, Oct 11 2015 *)

A237271(a(k)) = A000005(a(k)).

A251820 Numbers n for which the symmetric representation of sigma(n) has at least 3 parts, all having the same area.

Original entry on oeis.org

15, 5950

Offset: 1

Hartmut F. W. Hoft, Dec 09 2014

a(3) > 36000000.

Also intersection of A241558 and A241559 (minimum = maximum) minus the union of A238443 and A239929 (number of parts <= 2).

			The parts of the symmetric representations of sigma(15) and sigma(5950) are {8, 8, 8} and {4464, 4464, 4464}, respectively, so a(1) = 15 and a(2) = 5950.
From _Omar E. Pol_, Dec 09 2014: (Start)
Illustration of the symmetric representation of sigma(15) = 8 + 8 + 8 = 24 in the first quadrant:
.
.  _ _ _ _ _ _ _ _ 8
. |_ _ _ _ _ _ _ _|
.                 |
.                 |_ _
.                 |_  |_ 8
.                   |   |_
.                   |_ _  |
.                       |_|_ _ _ 8
.                             | |
.                             | |
.                             | |
.                             | |
.                             | |
.                             | |
.                             | |
.                             |_|
.
The three parts have the same area.
(End)

Cf. A000203, A237593, A237270, A238443, A239663, A241558, A241559.

(* T[], row[], cD[] & tD[] are defined in A239663 *)
a251820[n_] := Module[{pT = T[n, 1], cT, cL, cW = 0, cR = 0, sects = {}, j = 1, r = row[n], test = True}, While[test && j <= r, cT = T[n, j+1]; cL = pT - cT; cW += (-1)^(j+1) * tD[n, j]; If[cW == 0 && cR != 0, AppendTo[sects, cR]; cR = 0; If[Min[sects] != Max[sects], test = False], cR += cL * cW]; pT = cT; j++]; If[cW != 0, AppendTo[sects, 2 * cR - cW]]; Min[sects] == Max[sects] && Length[sects] > 1]
Select[Range[50000], a251820] (* data *)

A296513 a(n) is the smallest subpart of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 1, 4, 15, 3, 9, 6, 5, 7, 12, 1, 31, 9, 2, 10, 3, 5, 18, 12, 13, 5, 21, 6, 1, 15, 3, 16, 63, 7, 27, 3, 10, 19, 30, 8, 11, 21, 4, 22, 42, 1, 36, 24, 29, 7, 15, 10, 49, 27, 3, 8, 9, 11, 45, 30, 6, 31, 48, 5, 127, 9, 1, 34, 63, 13, 13, 36, 7, 37, 57, 3

Offset: 1

Omar E. Pol, Feb 10 2018

nonn

If n is an odd prime (A065091) then a(n) = (n + 1)/2.
If n is a power of 2 (A000079) then a(n) = 2*n - 1.
If n is a perfect number (A000396) then a(n) = 1 assuming there are no odd perfect numbers.
a(n) is also the smallest number in the n-th row of the triangles A279391 and A280851.
a(n) is also the smallest nonzero term in the n-th row of triangle A296508.
The symmetric representation of sigma(n) has A001227(n) subparts.
For the definition of the "subpart" see A279387.
For a diagram with the subparts for the first 16 positive integers see A296508.
It appears that a(n) = 1 if and only if n is a hexagonal number (A000384). - Omar E. Pol, Sep 08 2021
The above conjecture is true. See A280851 for a proof. - Omar E. Pol, Mar 10 2022

			For n = 15 the subparts of the symmetric representation of sigma(15) are [8, 7, 1, 8], the smallest subpart is 1, so a(15) = 1.

Shares infinitely many terms with A241558, A241559, A241838, A296512 (and possibly more).

Cf. A000079, A000203 (sum of subparts), A000225, A000384, A000396, A001227 (number of subparts), A065091, A099378, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A279387, A279391, A280850, A280851, A296508.

(* a280851[] and support function are defined in A280851 *)
a296513[n_]:=Min[a280851[n]]
Map[a296513,Range[75]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

More terms from Omar E. Pol, Aug 28 2021

A296512 a(n) is the largest subpart of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 11, 4, 15, 5, 9, 6, 23, 7, 12, 8, 31, 9, 35, 10, 39, 11, 18, 12, 47, 13, 21, 14, 55, 15, 59, 16, 63, 17, 27, 18, 71, 19, 30, 20, 79, 21, 83, 22, 42, 27, 36, 24, 95, 25, 39, 26, 49, 27, 107, 28, 111, 29, 45, 30, 119, 31, 48, 32, 127, 33, 131, 34, 63

Offset: 1

Omar E. Pol, Feb 10 2018

nonn

If n is an odd prime (A065091) then a(n) = (n + 1)/2.
If n is a power of 2 (A000079) then a(n) = 2*n - 1.
If n is a perfect number (A000396) then a(n) = 2*n - 1, assuming there are no odd perfect numbers.
a(n) is also the largest element in the n-th row of the triangles A279391, A280851 and A296508.
The symmetric representation of sigma(n) has A001227(n) subparts.
For the definition of the "subpart" see A279387.
For a diagram with the subparts for the first 16 positive integers see A296508.

			For n = 15 the subparts of the symmetric representation of sigma(15) are [8, 7, 1, 8], the largest subpart is 8, so a(15) = 8.

Shares infinitely many terms with A241558, A241559, A241838, A296513 (and possibly more).

Cf. A000079, A000203 (sum of subparts), A000225, A000396, A001227 (number of subparts), A065091, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A279387, A279391, A280850, A280851, A296508.

(* a280851[] and support function are defined in A280851 *)
a296512[n_]:=Max[a280851[n]]
Map[a296512,Range[68]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

More terms from Omar E. Pol, Aug 28 2021

A377462 a(n) is the size of the central part of the symmetric representation of sigma(n), or 0 if such a part does not exits.

Original entry on oeis.org

1, 3, 0, 7, 0, 12, 0, 15, 3, 0, 0, 28, 0, 0, 8, 31, 0, 39, 0, 42, 0, 0, 0, 60, 5, 0, 0, 56, 0, 72, 0, 63, 0, 0, 12, 91, 0, 0, 0, 90, 0, 96, 0, 0, 32, 0, 0, 124, 7, 15, 0, 0, 0, 120, 0, 120, 0, 0, 0, 168, 0, 0, 16, 127, 0, 144, 0, 0, 0, 36, 0, 195, 0, 0, 0, 0, 18, 0, 0, 186, 9, 0, 0, 224, 0

Offset: 1

Omar E. Pol, Oct 29 2024

nonn

a(n) = A000203(n) if and only if n is a member of A174973.

			For n = 9 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] and the central part is 3 so a(9) = 3.

Indices of odd terms give A028982.
Indices of even terms give A028983.
Indices of zeros give A071561.
Indices of nonzero terms give A071562.
Nonzero terms give A295423.
Parity gives A053866.
Has the same parity as A000203, A000593, A001227, A033879, A033880, A067742.
Cf. A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A241558, A241559, A245092, A262626, A347950.

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A241558 Smallest part of the symmetric representation of sigma(n).

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A241838 Column 1 of A237270, also the right border.

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A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

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A251820 Numbers n for which the symmetric representation of sigma(n) has at least 3 parts, all having the same area.

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A296513 a(n) is the smallest subpart of the symmetric representation of sigma(n).

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A296512 a(n) is the largest subpart of the symmetric representation of sigma(n).

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A377462 a(n) is the size of the central part of the symmetric representation of sigma(n), or 0 if such a part does not exits.

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