A241838 -id:A241838 - 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

A241559 Largest part of the symmetric representation of sigma(n).

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, 32, 36, 24, 124, 25, 39, 26, 49, 27, 120, 28, 120, 29, 45, 30, 168, 31, 48, 32, 127

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

First differs from A241838 at a(45).
If A237271(n) = 1 then a(n) = A241558(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241558(n) = A241838(n).
For more information see 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 largest part is 5 so a(9) = 5.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the largest part is 32 so a(45) = 32.

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241558, A071561, A071562, A241838.

(* Function a237270[] is defined in A237270 *)
a241559[n_]:=Max[a237270[n]]
Map[a241559,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

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

A241739 Number of partitions p of n such that (number of numbers in p of form 3k+1) > (number of numbers in p of form 3k+2).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 3, 8, 9, 10, 20, 24, 27, 49, 58, 69, 109, 132, 153, 234, 279, 331, 469, 565, 662, 918, 1093, 1290, 1723, 2056, 2411, 3165, 3751, 4411, 5656, 6700, 7839, 9932, 11707, 13699, 17084, 20099, 23441, 28939, 33914, 39498, 48236, 56392, 65481

Offset: 0

Clark Kimberling, Apr 28 2014

Each number in p is counted once, regardless of its multiplicity.

			a(8) counts these 9 partitions: 71, 611, 44, 431, 4211, 3311, 311111, 11111111.

Cf. A241737, A241738, A241740, A241743.

z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[1, p] < s[2, p]], {n, 0, z}]  (* A241737 *)
Table[Count[f[n], p_ /; s[1, p] == s[2, p]], {n, 0, z}] (* A241738 *)
Table[Count[f[n], p_ /; s[1, p] > s[2, p]], {n, 0, z}]  (* A241739 *)

a(n) + A241737(n) + A241838(n) = A000041(n) for n >= 0.

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

A299778 Irregular triangle read by rows: T(n,k) is the part that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned part is already associated to a previous peak or if there is no part adjacent to the k-th peak, with n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 0, 4, 15, 0, 0, 5, 3, 5, 9, 0, 9, 0, 6, 0, 0, 6, 28, 0, 0, 0, 7, 0, 0, 7, 12, 0, 12, 0, 8, 8, 0, 0, 8, 31, 0, 0, 0, 0, 9, 0, 0, 0, 9, 39, 0, 0, 0, 0, 10, 0, 0, 0, 10, 42, 0, 0, 0, 0, 11, 5, 0, 5, 0, 11, 18, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 12, 60, 0, 0, 0, 0, 0, 13, 0, 5, 0, 0, 13

Offset: 1

Omar E. Pol, Apr 03 2018

For the definition of "part" of the symmetric representation of sigma see A237270.

For more information about the mentioned Dyck paths see A237593.

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  0,  4;
  15,  0,  0;
   5,  3,  5;
   9,  0,  9,  0;
   6,  0,  0,  6;
  28,  0,  0,  0;
   7,  0,  0,  7;
  12,  0, 12,  0;
   8,  8,  0,  0,  8;
  31,  0,  0,  0,  0;
   9,  0,  0,  0,  9;
  39,  0,  0,  0,  0;
  10,  0,  0,  0, 10;
  42,  0,  0,  0,  0;
  11,  5,  0,  5,  0, 11;
  18,  0,  0,  0, 18,  0;
  12,  0,  0,  0,  0, 12;
  60,  0,  0,  0,  0,  0;
  13,  0,  5,  0,  0, 13;
  21,  0,  0,  0  21,  0;
  14,  6,  0,  6,  0, 14;
  56,  0,  0,  0,  0,  0,  0;
  ...
Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.              0 _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _ 0
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_ 0
.    0 _ _ _| |    0 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |    0 _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|  0 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |    0 _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|  0 _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|0    | |     | |
.   | |     |_|_ _     0  |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|0    | |
.   |_|_ _ _     0  |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _   0  |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|  0 _| |      0
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |    0 _|  _|
.          0  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|  0
.            0    |                             28|  _ _|  0
.                 |_ _ _ _ _ _ _ _                | |    0
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
For the construction of the spiral see A239660.

Index entries for sequences related to sigma(n)

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
Nonzero terms give A237270.
The number of nonzero terms in row n is A237271(n).
Column 1 is A241838.
The triangle with n rows contain A237590(n) nonzero terms.
Cf. A296508 (analog for subparts).
Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850, A280851.

A339583 Leading term in row 2*n of A237270.

Original entry on oeis.org

3, 7, 12, 15, 9, 28, 12, 31, 39, 42, 18, 60, 21, 56, 72, 63, 27, 91, 30, 90, 96, 42, 36, 124, 39, 49, 120, 120, 45, 168, 48, 127, 144, 63, 54, 195, 57, 70, 84, 186, 63, 224, 66, 180, 234, 84, 72, 252, 75, 217, 108, 210, 81, 280, 84, 248, 120, 105, 90, 360, 93, 112, 312, 255, 99, 336

Offset: 1

N. J. A. Sloane, Dec 25 2020

nonn

The leading term in row 2*n-1 is n.
The first column in A237270, [1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, ...], without the initial 1, is A237270(A237590(n)+1). By bisecting this we get the present sequence.
Bisection of A241838. - Omar E. Pol, Feb 23 2021

			Row 10 of A237270 is [9, 9], so a(5) = 9 (the first of the two 9's, officially).

N. J. A. Sloane, Table of n, a(n) for n = 1..2495

Cf. A237270, A237271, A237590, A237593, A241838.

A302248 Irregular triangle read by rows in which the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A299778.

Original entry on oeis.org

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 4, 0, 15, 0, 0, 5, 5, 3, 9, 0, 0, 9, 6, 6, 0, 0, 28, 0, 0, 0, 7, 7, 0, 0, 12, 0, 0, 12, 8, 8, 8, 0, 0, 31, 0, 0, 0, 0, 9, 9, 0, 0, 0, 39, 0, 0, 0, 0, 10, 10, 0, 0, 0, 42, 0, 0, 0, 0, 11, 11, 5, 0, 0, 5, 18, 0, 0, 18, 0, 0, 12, 12, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 13, 13, 0, 0, 5, 0

Offset: 1

Omar E. Pol, Apr 10 2018

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  4,  0;
  15,  0,  0;
   5,  5,  3;
   9,  0,  0,  9;
   6,  6,  0,  0;
  28,  0,  0,  0;
   7,  7,  0,  0;
  12,  0,  0, 12;
   8,  8,  8,  0,  0;
  31,  0,  0,  0,  0;
   9,  9,  0,  0,  0;
  39,  0,  0,  0,  0;
  10, 10,  0,  0,  0;
  42,  0,  0,  0,  0;
  11, 11,  5,  0,  0,  5;
  18,  0,  0, 18,  0,  0;
  12, 12,  0,  0,  0,  0;
  60,  0,  0,  0,  0,  0;
  13, 13,  0,  0,  5,  0;
  21,  0,  0, 21,  0,  0;
  14, 14,  6,  0,  0,  6;
  56,  0,  0,  0,  0,  0,  0;
...
For n = 21 the 21st row of A299778 is [11, 5, 0, 5, 0, 11], so the 21st row of this triangle is [11, 11, 5, 0, 0, 5].

Index entries for sequences related to sigma(n)

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
The number of nonzero terms in row n is A237271(n).
Column 1 is A241838.
The triangle with n rows contain A237590(n) nonzero terms.
Cf. A280850 (analog for subparts).
Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239660, A240542, A244050, A245092, A262626, A279387, A279388, A280851, A299778.

cp's OEIS Frontend

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

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A241559 Largest part of the symmetric representation of sigma(n).

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

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A241739 Number of partitions p of n such that (number of numbers in p of form 3k+1) > (number of numbers in p of form 3k+2).

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

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A339583 Leading term in row 2*n of A237270.

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A302248 Irregular triangle read by rows in which the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A299778.

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