A007368 -id:A007368 - cp's OEIS frontend

A201915 Each row of triangle T(n,k) has a sorted list of n values such that sigma(T(n,k)) = A007368(n).

1, 6, 11, 14, 15, 23, 42, 62, 69, 77, 30, 46, 51, 55, 71, 60, 78, 92, 123, 143, 167, 114, 135, 158, 177, 203, 209, 239, 132, 140, 182, 188, 195, 249, 287, 299, 120, 174, 184, 190, 267, 295, 319, 323, 359, 204, 220, 224, 246, 284, 286, 334, 415, 451, 503

Offset: 1

T. D. Noe, Jan 24 2012

The first and last terms of each row are given in A184393 and A184394.

Note that the integers in the 10th row have sigma(n)=504 (A180164(1)) and thus include A002025(1) and A002046(1). - Michel Marcus, Oct 22 2013

			Triangle:
1
6,    11
14,   15,  23
42,   62,  69,  77
30,   46,  51,  55,  71
60,   78,  92, 123, 143, 167
114, 135, 158, 177, 203, 209, 239
132, 140, 182, 188, 195, 249, 287, 299
120, 174, 184, 190, 267, 295, 319, 323, 359
204, 220, 224, 246, 284, 286, 334, 415, 451, 503

T. D. Noe, Rows n = 1..100 of triangle, flattened

A184393 The smallest number m such that sigma(m) = A007368(n), where A007368(n) = the smallest k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

1, 6, 14, 42, 30, 60, 114, 132, 120, 204, 210, 480, 408, 390, 264, 930, 1080, 1860, 864, 870, 552, 3120, 4080, 1140, 1380, 1020, 2460, 2184, 840, 2040, 3480, 4140, 1560, 2208, 1320, 3780, 1848, 5544, 7590, 6468, 8544, 13500, 8280, 8190, 4872, 4620, 8856

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

A184394 is the sequence of the largest such numbers. Row n of A201915 gives all n values satisfying sigma(x) = A007368(n).

			For n = 5, sequence of defined numbers m_5: 30, 46, 51, 55, 71; a(5) = 30.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Extended by T. D. Noe, Jan 24 2012

A184394 The largest number m such that sigma(m) = A007368(n), where A007368(n) = the smallest k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

1, 11, 23, 77, 71, 167, 239, 299, 359, 503, 527, 1511, 1007, 943, 719, 2201, 3427, 5207, 2419, 2059, 1439, 10187, 12811, 3359, 3901, 3023, 6887, 6719, 2879, 6319, 10799, 13103, 5039, 6047, 4189, 13193, 5609, 18719, 20437, 18871, 22679, 43259, 27707, 25853

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

A184393(n) is the sequence of the smallest such numbers. Row n of A201915 gives all n values satisfying sigma(x) = A007368(n).

			For n = 5, sequence of defined numbers m_5: 30, 46, 51, 55, 71; a(5) = 71.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Extended by T. D. Noe, Jan 24 2012

A061081 Duplicate of A007368.

Original entry on oeis.org

2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600

Offset: 0

dead

A090554 Erroneous version of A007368.

Original entry on oeis.org

1, 12, 42, 96, 72

Offset: 1

dead

A054973 Number of numbers whose divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 3, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 4, 0

Offset: 1

Henry Bottomley, May 16 2000

nonn

a(n) = frequency of values n in A000203(m), where A000203(m) = sum of divisors of m. a(n) >= 1 for such n that A175192(n) = 1, a(n) >= 1 if A000203(m) = n for any m. a(n) = 0 for such n that A175192(n) = 0, a(n) = 0 if A000203(m) = n has no solution. - Jaroslav Krizek, Mar 01 2010
First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368. - Robert G. Wilson v, May 14 2014
a(n) is also the number of positive terms in the n-th row of triangle A299762. - Omar E. Pol, Mar 14 2018
Also the number of integer partitions of n whose parts form the set of divisors of some number (necessarily the greatest part). The Heinz numbers of these partitions are given by A371283. For example, the a(24) = 3 partitions are: (23,1), (15,5,3,1), (14,7,2,1). - Gus Wiseman, Mar 22 2024

			a(12) = 2 since 11 has factors 1 and 11 with 1 + 11 = 12 and 6 has factors 1, 2, 3 and 6 with 1 + 2 + 3 + 6 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sum-of-divisors function).
For partial sums see A074753.
Cf. A002191, A007609, A308039.
The non-strict version is A371284, ranks A371288.
These partitions have ranks A371283, unsorted version A275700.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
Cf. A001221, A002865, A008289, A371286, A371285.

nn = 105; t = Table[0, {nn}]; k = 1; While[k < 6 nn^(3/2)/Pi^2, d = DivisorSigma[1, k]; If[d < nn + 1, t[[d]]++]; k++]; t (* Robert G. Wilson v, May 14 2014 *)
Table[Length[Select[IntegerPartitions[n],#==Reverse[Divisors[Max@@#]]&]],{n,30}] (* Gus Wiseman, Mar 22 2024 *)

a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i));); #v; \\ Michel Marcus, Oct 22 2013

a(n)=sum(k=1,n,sigma(k)==n) \\ Charles R Greathouse IV, Nov 12 2013

first(n)=my(v=vector(n),t); for(k=1,n, t=sigma(n); if(t<=n, v[t]++)); v \\ Charles R Greathouse IV, Mar 08 2017

A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308039. - Amiram Eldar, Dec 23 2024

Incorrect comment deleted by M. F. Hasler, Nov 21 2019

A007609 Values taken by the sigma function A000203, listed with multiplicity and in ascending order.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 12, 13, 14, 15, 18, 18, 20, 24, 24, 24, 28, 30, 31, 31, 32, 32, 36, 38, 39, 40, 42, 42, 42, 44, 48, 48, 48, 54, 54, 56, 56, 57, 60, 60, 60, 62, 63, 68, 72, 72, 72, 72, 72, 74, 78, 80, 80, 84, 84, 84, 90, 90, 90, 91, 93, 96, 96, 96, 96, 98, 98

Offset: 1

Walter Nissen

A175192(a(n)) = 1, A054973(a(n)) >= 1. - Jaroslav Krizek, Mar 01 2010

a(n) is the median of the values of A000203(m) from m=1 to m=2n-1. (This needs confirmation as it relies on the assumption that A000203(n) is always bigger than the median of the values A000203(x) from x=1 to x=n.) - Chayim Lowen, May 27 2015

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

Cf. A000203, A002191 (duplicates removed), A007368, A085790.

sort(select(`<=`,map(numtheory:-sigma,[$1..1000]),1001)); # Robert Israel, Jun 04 2015

terms = 68; ClearAll[t]; t[k_] := t[k] = Sort[ Table[ DivisorSigma[1, n], {n, 1, k*terms}]][[1 ;; terms]]; t[k = 2]; While[t[k] != t[k-1], k++]; t[k] (* Jean-François Alcover, Nov 21 2012 *)
With[{nn=80},Take[Sort[DivisorSigma[1,Range[nn*100]]],nn]] (* Harvey P. Dale, Mar 09 2016 *)

list(lim)=select(k->k<=lim,Set(apply(sigma,[1..lim\1]))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A085790(n)). - Jinyuan Wang, Apr 15 2020

A145899 Numbers n such that sigma(x) = n has more solutions x than any smaller n.

Original entry on oeis.org

1, 12, 24, 72, 168, 240, 336, 360, 504, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 15120, 17280, 20160, 30240, 40320, 60480, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 1088640, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800

Offset: 1

Douglas E. Iannucci, Oct 22 2008

nonn

Sequence A206027 has the number of solutions.

			sigma(m)=1 has only one solution: m=1.
sigma(m)=12 has two solutions, m=6 and m=11; 12 is the smallest number with more than one such solutions.
sigma(m)=24 has three solutions, m=14,m=15 and m=23; 24 is the smallest number with more than two such solutions.
sigma(m)=72 has five solutions, m=30, m=46, m=51, m=55 and m=71; 72 is the smallest number with more than three such solutions.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000203 (sum of divisors of n), A054973 (number of numbers whose divisors sum to n), A007368 (smallest k such that sigma(x) = k has exactly n solutions).
Cf. A206027.
Cf. Untouchable numbers (A005114), sigma-untouchable numbers (A007369) and highly touchable numbers (A238895).

t = DivisorSigma[1, Range[10^6]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[1]]]], {n, Length[t2]}]; t3 (* T. D. Noe, Feb 03 2012 *)

{m=3650000; v=vectorsmall(m); for(n=1, m, s=sigma(n); if(s<=m, v[s]++)); g=0; j=1; while(j<=m, if(v[j]<=g, j++, g=v[j]; print1(j, ",")))} \\ Klaus Brockhaus, Oct 27 2008

Extended beyond a(15) by Klaus Brockhaus, Oct 27 2008

A159886 Values k such that sigma(x) = k has more than one solution, sigma = A000203.

Original entry on oeis.org

12, 18, 24, 31, 32, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 98, 104, 108, 114, 120, 124, 126, 128, 132, 140, 144, 152, 156, 168, 180, 182, 186, 192, 210, 216, 224, 228, 234, 240, 248, 252, 264, 270, 272, 280, 288, 294, 308, 312, 320, 324, 336, 342, 360, 372, 378, 384, 390

Offset: 1

Jaroslav Krizek, Apr 25 2009

nonn

Numbers k with A054973(k) >= 2. Numbers k which occur in A000203 more than once.
Numbers k = A007609(n) with A007609(n+1) - A007609(n) = 0.
Does this sequence have finite density? - Franklin T. Adams-Watters, Jun 18 2009
See A300869 for the odd terms, much less frequent since they can only occur for x = k^2 or 2*k^2. - M. F. Hasler, Mar 16 2018

			a(1) = 12 as the multiplicity of the value 12 is 2: 12 = sigma(6) = sigma(11).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1095 from Franklin T. Adams-Watters)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A007609, A007368.
Subsequence of A002191.
Odd terms are listed in A300869.

na(n) = local(v, s); v=vector(n);for(k=1,n,s=sigma(k);if(s<=n,v[s]++));v
la(n) = local(v, r); v=na(n);r=[];for(k=1,n,if(v[k]>1,r=concat(r,[k])));r \\ Franklin T. Adams-Watters, Jun 18 2009

is(k) = invsigmaNum(k) > 1; \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

Edited and extended by R. J. Mathar, Apr 28 2009

A299762 Irregular triangle T(n,k) read by rows in which row n lists the positive integers whose sum of divisors is n, or 0 if no such integer exists.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 11, 9, 13, 8, 0, 0, 10, 17, 0, 19, 0, 0, 0, 14, 15, 23, 0, 0, 0, 12, 0, 29, 16, 25, 21, 31, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 26, 41, 0, 43, 0, 0, 0, 33, 35, 47, 0, 0, 0, 0, 0, 34, 53, 0, 28, 39, 49, 0, 0, 24, 38, 59, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 46, 51, 55, 71, 0, 73

Offset: 1

Omar E. Pol, Mar 12 2018

Essentially the same as the triangle described in the example section of A085790, but with 0's added in empty rows.

Are the records the same as A008578?

			First 24 rows of triangle T(n,k):
-----------------------
. n / k:  1   2   3 ...
-----------------------
| 1|      1;
| 2|      0;
| 3|      2;
| 4|      3;
| 5|      0;
| 6|      5;
| 7|      4;
| 8|      7;
| 9|      0;
|10|      0;
|11|      0;
|12|      6, 11;
|13|      9;
|14|     13;
|15|      8;
|16|      0;
|17|      0;
|18|     10, 17;
|19|      0;
|20|     19;
|21|      0;
|22|      0;
|23|      0;
|24|     14, 15, 23;
...
For n = 23 there are no positive integers whose sum of divisors is 23, so T(23, 1) = 0, which is the only element in the 23rd row of the triangle.
For n = 24 there are three positive integers whose sum of divisors is 24; they are 14, 15 and 23, since sigma(14) = 1 + 2 + 7 + 14 = 24, sigma(15) = 1 + 3 + 5 + 15 = 24 and sigma(23) = 1 + 23 = 24, so the 24th row of the triangle is [14, 15, 23].

Row sums give A258913.
Column 1 gives A051444.
Right border gives A057637.
Positive terms give A085790.
Row n has A054973(n) positive integers.
Positive terms in the first column give A002192.
Indices of the rows that contain a zero give A007369.
Indices of the rows that contain positive terms give A002191.
Cf. A000203, A007368, A007609, A008578.

With[{nn = 74}, ReplacePart[ConstantArray[{0}, nn], PositionIndex@ Array[DivisorSigma[1, #] &, nn]]] // Flatten (* Michael De Vlieger, Mar 16 2018 *)

sigma(T(n,k)) = n, if T(n,k) >= 1.

cp's OEIS Frontend

A201915 Each row of triangle T(n,k) has a sorted list of n values such that sigma(T(n,k)) = A007368(n).

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A184393 The smallest number m such that sigma(m) = A007368(n), where A007368(n) = the smallest k such that sigma(x) = k has exactly n solutions.

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A184394 The largest number m such that sigma(m) = A007368(n), where A007368(n) = the smallest k such that sigma(x) = k has exactly n solutions.

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A061081 Duplicate of A007368.

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A090554 Erroneous version of A007368.

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A054973 Number of numbers whose divisors sum to n.

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A007609 Values taken by the sigma function A000203, listed with multiplicity and in ascending order.

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A145899 Numbers n such that sigma(x) = n has more solutions x than any smaller n.

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A159886 Values k such that sigma(x) = k has more than one solution, sigma = A000203.

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