A053224 -id:A053224 - cp's OEIS frontend

A224613 a(n) = sigma(6*n).

12, 28, 39, 60, 72, 91, 96, 124, 120, 168, 144, 195, 168, 224, 234, 252, 216, 280, 240, 360, 312, 336, 288, 403, 372, 392, 363, 480, 360, 546, 384, 508, 468, 504, 576, 600, 456, 560, 546, 744, 504, 728, 528, 720, 720, 672, 576, 819, 684, 868, 702, 840, 648

Offset: 1

Zak Seidov, Apr 22 2013

nonn

Conjectures: sigma(6n) > sigma(6n - 1) and sigma(6n) > sigma(6n + 1).
Conjectures are false. Try prime 73961483429 for n. One finds sigma(6*73961483429) < sigma(6*73961483429+1). The number n = 105851369791 provides a counterexample for the other case. - T. D. Noe, Apr 22 2013
Sum of the divisors of the numbers k which have the property that the width associated to the vertex of the first (also the last) valley of the smallest Dyck path of the symmetric representation of sigma(k) is equal to 2 (see example). Other positive integers have width 0 or 1 associated to the mentioned valley. - Omar E. Pol, Aug 11 2021

			From _Omar E. Pol_, Aug 11 2021: (Start)
Illustration of initial terms:
----------------------------------------------------------------------
   n    6*n   a(n)    Diagram:  1           2           3           4
----------------------------------------------------------------------
                                _           _           _           _
                               | |         | |         | |         | |
                               | |         | |         | |         | |
                          * _ _| |         | |         | |         | |
                           |  _ _|         | |         | |         | |
                      _ _ _| |_|           | |         | |         | |
   1     6     12    |_ _ _ _|      * _ _ _| |         | |         | |
                                    _|  _ _ _|         | |         | |
                                * _|  _| |             | |         | |
                                 |  _|  _|    * _ _ _ _| |         | |
                                 | |_ _|       |  _ _ _ _|         | |
                      _ _ _ _ _ _| |          _| | |               | |
   2    12     28    |_ _ _ _ _ _ _|        _|  _|_|    * _ _ _ _ _| |
                                      * _ _|  _|         |  _ _ _ _ _|
                                       |  _ _|        _ _| | |
                                       | |_ _|      _|  _ _| |
                                       | |        _|  _|  _ _|
                      _ _ _ _ _ _ _ _ _| |       |  _|  _|
   3    18     39    |_ _ _ _ _ _ _ _ _ _|  * _ _| |  _|
                                             |  _ _| |
                                             | |_ _ _|
                                             | |
                                             | |
                      _ _ _ _ _ _ _ _ _ _ _ _| |
   4    24     60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the mentioned vertices are aligned on two straight lines that meet at point (3,3).
a(n) equals the area (also the number of cells) in the n-th diagram. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), this sequence (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
Cf. A000203 (sigma(n)), A053224 (n: sigma(n) < sigma(n+1)).
Cf. A067825 (even n: sigma(n)< sigma(n+1)).
Cf. A008588, A237593.

DivisorSigma[1,6*Range[60]] (* Harvey P. Dale, Apr 16 2016 *)

a(n)=sigma(6*n) \\ Charles R Greathouse IV, Apr 22 2013

from sympy import divisor_sigma
def a(n):  return divisor_sigma(6*n)
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Dec 28 2021

from math import prod
from collections import Counter
from sympy import factorint
def A224613(n): return prod((p**(e+1)-1)//(p-1) for p, e in (Counter(factorint(n))+Counter([2,3])).items()) # Chai Wah Wu, Sep 07 2023

a(n) = A000203(6n).
a(n) = A000203(A008588(n)). - Omar E. Pol, Aug 11 2021
Sum_{k=1..n} a(k) = (55*Pi^2/72) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

Corrected by Harvey P. Dale, Apr 16 2016

A053226 Numbers k for which sigma(k) > sigma(k+1).

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 117, 118, 120, 122, 124, 126, 128, 130

Offset: 1

Asher Auel, Jan 06 2000

nonn

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A000203, A053222, A053224, A053238.

import Data.List (findIndices)
a053226 n = a053226_list !! (n-1)
a053226_list = map (+ 1) $ findIndices (< 0) a053222_list
-- Reinhard Zumkeller, Oct 16 2011

with(numtheory): seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..139);

Select[Range[150], DivisorSigma[1, #] > DivisorSigma[1, # + 1] &] (* Carl Najafi, Aug 16 2011 *)

is(n)=sigma(n)>sigma(n+1) \\ Charles R Greathouse IV, Mar 09 2014

A053222 First differences of sigma(n).

Original entry on oeis.org

2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8, 23, -43, 60, -76, 58

Offset: 1

Asher Auel, Jan 06 2000

a(A002961(n)) = 0. - Reinhard Zumkeller, Dec 28 2011

Considering the values |a(n)| <= 100 for n < 10^13, we notice that some odd values do not appear within that range, namely 9, 17, 25, 27, 33, 37, 39, 45, 47, 49, 51, 55, 57, 59, 69, 71, 77, 81, 83, 87, 89, 91, 95, 97, and 99. All the other absolute values <= 100 appear for n < 3600, with the exception of a(1159742043) = 62. - Giovanni Resta, Jun 26 2017

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A053223, A053225, A053227, A053224, A053226.

List([1..70], n -> Sigma(n+1)-Sigma(n)); # Muniru A Asiru, Feb 14 2018

a053222 n = a053222_list !! (n-1)
a053222_list = zipWith (-) (tail a000203_list) a000203_list
-- Reinhard Zumkeller, Oct 16 2011

[DivisorSigma(1, n+1) - DivisorSigma(1,n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

A053222 := proc(n)
numtheory[sigma](n+1)-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Jul 08 2013

DivisorSigma[1, Range[100]] // Differences (* Jean-François Alcover, Jan 26 2018 *)

a(n)=sigma(n+1)-sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = A000203(n+1) - A000203(n).
G.f.:  2*(x-1)/(Q(0) -  2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+3)*(k+1)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
G.f.: -1 + (1 - x)*Sum_{k>=1} k*x^(k-1)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A053230 First differences between numbers k for which sigma(k) < sigma(k+1).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Asher Auel, Jan 10 2000

It seems that the expansion consists of only {1,2,3,4}.
The first exception is a(10010491) = 6, corresponding to the gap from 20021153 to 20021159. - Charles R Greathouse IV, Mar 09 2014
The asymptotic mean of this sequence is 2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A000203, A053224, A053231, A053232, A053233, A053234, A053235, A053236, A053237, A053238.

a053230 n = a053230_list !! (n-1)
a053230_list = zipWith (-) (tail a053224_list) a053224_list
-- Reinhard Zumkeller, May 07 2012

with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( f[i+1] - f[i], i=1..2000);

Differences[Select[Range[250],DivisorSigma[1,#]Harvey P. Dale, Apr 30 2011 *)

last=ls=1; for(n=2,200,ns=sigma(n+1); if(ls>=ns,ls=ns; next); ls=ns; print1(n-last", ");last=n) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = A053224(n+1) - A053224(n).

A053233 Numbers n such that A053230(n) = 2.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 85

Offset: 1

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A053224, A053230, A053231, A053232, A053234, A053235, A053236, A053237.

import Data.List (elemIndices)
a053233 n = a053233_list !! (n-1)
a053233_list = map (+ 1) $ elemIndices 2 a053230_list
-- Reinhard Zumkeller, May 07 2012

with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 2,i,print( )), i=1..100);

Position[Differences@ Select[Range[170], Less @@ DivisorSigma[1, # + {0, 1}] &], 2][[All, 1]] (* Michael De Vlieger, Nov 19 2019 *)

A053234 Numbers n such that A053230(n) = 1.

Original entry on oeis.org

1, 2, 31, 32, 38, 39, 67, 68, 74, 75, 98, 99, 128, 129, 157, 197, 198, 201, 228, 229, 240, 241, 247, 248, 262, 277, 278, 283, 284, 307, 308, 313, 314, 332, 333, 339, 340, 349, 369, 370, 382, 383, 386, 400, 401, 413, 414, 430, 431, 459, 460, 475, 489, 490, 502

Offset: 1

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053235, A053236, A053237.

import Data.List (elemIndices)
a053234 n = a053234_list !! (n-1)
a053234_list = map (+ 1) $ elemIndices 1 a053230_list
-- Reinhard Zumkeller, May 07 2012

with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 2,i,print( )), i=1..1000);

A053235 Numbers n such that A053230(n) = 3.

Original entry on oeis.org

158, 202, 263, 350, 387, 476, 567, 582, 701, 790, 879, 894, 926, 999, 1103, 1236, 1282, 1403, 1418, 1501, 1523, 1646, 1661, 1737, 1831, 1847, 1953, 2059, 2074, 2149, 2185, 2237, 2265, 2370, 2505, 2563, 2683, 2729, 2873, 2894, 2909, 3032, 3107, 3127

Offset: 1

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053236, A053237.

import Data.List (elemIndices)
a053235 n = a053235_list !! (n-1)
a053235_list = map (+ 1) $ elemIndices 3 a053230_list
-- Reinhard Zumkeller, May 07 2012

with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 3,i,print( )), i=1..1000);

Position[Differences@ Select[Range[10^4], Less @@ DivisorSigma[1, # + {0, 1}] &], 3][[All, 1]] (* Michael De Vlieger, Nov 19 2019 *)

A053236 Numbers n such that A053230(n) = 4.

Original entry on oeis.org

23, 54, 59, 84, 114, 138, 149, 172, 177, 232, 257, 281, 293, 311, 355, 392, 417, 422, 434, 445, 481, 506, 561, 596, 601, 644, 656, 686, 715, 745, 763, 775, 798, 809, 853, 864, 944, 955, 979, 984, 1013, 1018, 1061, 1072, 1140, 1164, 1187, 1192, 1222, 1227

Offset: 1

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053235, A053237.

import Data.List (elemIndices)
a053236 n = a053236_list !! (n-1)
a053236_list = map (+ 1) $ elemIndices 4 a053230_list
-- Reinhard Zumkeller, May 07 2012

with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..8000)];
seq( `if`(f[i+1] - f[i] = 4,i,print( )), i=1..3000);

A053231 First differences between n for which sigma(n) < sigma(n+1), which are not 2.

Original entry on oeis.org

1, 1, 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 4, 1, 3, 4, 4, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 4, 1, 3, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 4, 4, 1, 1, 4, 4, 1, 1, 1, 3, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 3, 1, 3, 4, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1

Offset: 1

Asher Auel, Jan 10 2000

nonn

Cf. A000203, A053224, A053230, A053232, A053233, A053234, A053235, A053236, A053237.

with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] <> 2,f[i+1] - f[i],print( )), i=1..2000);

The expansion of A053230 excluding 2's. a(n) = A053230(A053232(n)).

cp's OEIS Frontend

A224613 a(n) = sigma(6*n).

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A053226 Numbers k for which sigma(k) > sigma(k+1).

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A053222 First differences of sigma(n).

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A340793 Sequence whose partial sums give A000203.

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A053230 First differences between numbers k for which sigma(k) < sigma(k+1).

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A053233 Numbers n such that A053230(n) = 2.

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A053234 Numbers n such that A053230(n) = 1.

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