A299762 -id:A299762 - cp's OEIS frontend

A054973 Number of numbers whose divisors sum to n.

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

A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 6, 9, 13, 8, 0, 0, 10, 0, 19, 0, 0, 0, 14, 0, 0, 0, 12, 0, 29, 16, 21, 0, 0, 0, 22, 0, 37, 18, 27, 0, 20, 0, 43, 0, 0, 0, 33, 0, 0, 0, 0, 0, 34, 0, 28, 49, 0, 0, 24, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 30, 0, 73, 0, 0, 0, 45, 0, 57, 0, 0, 0, 44, 0, 0, 0, 0, 0

Offset: 1

Jud McCranie

Column 1 of A299762. - Omar E. Pol, Mar 14 2018

This is a right inverse of sigma = A000203 on A002191 = range(sigma): if n is in A002191, then there is some x with sigma(x) = n, and by definition a(n) is the smallest such x, so sigma(a(n)) = n. - M. F. Hasler, Nov 22 2019

			sigma(1) = 1, so a(1) = 1.
There is no k with sigma(k) = 2, since sigma(k) >= k + 1 for all k > 1 and sigma(1) = 1, so a(2) = 0.
sigma(4) = 7, and 4 is the smallest (since only) such number, so a(7) = 4.
6 and 12 are the only k with sigma(k) = 12, so 6 is the smallest and a(12) = 6.

R. K. Guy, Unsolved Problems Theory of Numbers, B1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005

Cf. A000203, A002192, A007626, A007369 (positions of zeros), A299762.

Do[ k = 1; While[ DivisorSigma[ 1, k ] != n && k < 10^4, k++ ]; If[ k != 10^4, Print[ k ], Print[ 0 ] ], {n, 1, 100} ]

a(n)=for(k=1,n,if(sigma(k)==n,return(k))); 0 \\ Charles R Greathouse IV, Mar 09 2014

A051444(n)=if(n=invsigma(n),vecmin(n)) \\ See Alekseyev link for invsigma(). An update including invsigmaMin = A051444 is planned. - M. F. Hasler, Nov 21 2019

Edited by M. F. Hasler, Nov 22 2019

A085790 Integers sorted by the sum of their divisors.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 9, 13, 8, 10, 17, 19, 14, 15, 23, 12, 29, 16, 25, 21, 31, 22, 37, 18, 27, 20, 26, 41, 43, 33, 35, 47, 34, 53, 28, 39, 49, 24, 38, 59, 61, 32, 67, 30, 46, 51, 55, 71, 73, 45, 57, 79, 44, 65, 83, 40, 58, 89, 36, 50, 42, 62, 69, 77, 52, 97, 101, 63, 103, 85

Offset: 1

Hugo Pfoertner, Jul 23 2003

Integers having the same sum of divisors are sorted in ascending order, e.g., sigma(14)=sigma(15)=sigma(23)=24 -> a(15)=14, a(16)=15, a(17)=23.
Also an irregular triangle where the k-th row consists of all numbers with divisor sum k. See A054973(k) for the k-th row length. - Jeppe Stig Nielsen, Jan 29 2015
By definition this is a permutation of the positive integers. Also positive integers of A299762. - Omar E. Pol, Mar 14 2018

			a(9) = 9, a(10) = 13, a(11) = 8 because sigma(9) = 9 + 3 + 1 = 13, sigma(13) = 13 + 1 = 14, sigma(8) = 8 + 4 + 2 + 1 = 15 and there are no other numbers with those sigma values.
Irregular triangle starts: (row numbers to the left are not part of the sequence)
   n : row(n)
   1 : 1,
   2 :
   3 : 2,
   4 : 3,
   5 :
   6 : 5,
   7 : 4,
   8 : 7,
   9 :
  10 :
  11 :
  12 : 6, 11,
  13 : 9,
  14 : 13,
  15 : 8,
  16 :
  17 :
  18 : 10, 17,
  19 :
  20 : 19,
  21 :
  22 :
  23 :
  24 : 14, 15, 23,
  25 :
- _Jeppe Stig Nielsen_, Feb 02 2015, edited by _M. F. Hasler_, Nov 21 2019

Hugo Pfoertner, Table of n, a(n) for n=1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005
Jeppe Stig Nielsen, First 10000 rows of the triangle for a(n)
Index entries for sequences that are permutations of the natural numbers

Cf. A000203 (sigma), A007609 (values taken by sigma, with multiplicity), A002191 (possible values for sigma), A002192 (first column).

Cf. A152454 (similar sequence for proper divisors only (aliquot parts)).

SortBy[Table[{n,DivisorSigma[1,n]},{n,120}],Last][[;;,1]] (* Harvey P. Dale, Sep 10 2024 *)

A085790_row(n)=invsigma(n) \\ Cf. Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

A057637 a(n) is the largest number k such that sigma(k) = n, where sigma is the sum of divisors function A000203, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 11, 9, 13, 8, 0, 0, 17, 0, 19, 0, 0, 0, 23, 0, 0, 0, 12, 0, 29, 25, 31, 0, 0, 0, 22, 0, 37, 18, 27, 0, 41, 0, 43, 0, 0, 0, 47, 0, 0, 0, 0, 0, 53, 0, 39, 49, 0, 0, 59, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 45, 0, 79, 0, 0, 0, 83, 0, 0, 0, 0, 0, 89

Offset: 1

Jud McCranie, Oct 10 2000

nonn

Right border of A299762. - Omar E. Pol, Mar 14 2018

			11 is the largest k such that sigma(k) = 12, so a(12) = 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A051444, A054973, A299762.

a[n_] := Module[{k = n}, While[k > 0 && DivisorSigma[1, k] != n, k--]; k]; Array[a, 90] (* Amiram Eldar, Jan 05 2020 *)

A057637(n)=if(n=A085790_row(n), n[#n]) \\ M. F. Hasler, Sep 21 2022

A258913 a(n) is the sum of all numbers k for which sigma(k) = n.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 17, 9, 13, 8, 0, 0, 27, 0, 19, 0, 0, 0, 52, 0, 0, 0, 12, 0, 29, 41, 52, 0, 0, 0, 22, 0, 37, 18, 27, 0, 87, 0, 43, 0, 0, 0, 115, 0, 0, 0, 0, 0, 87, 0, 67, 49, 0, 0, 121, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 253, 0, 73, 0, 0, 0

Offset: 1

Jeppe Stig Nielsen, Jun 14 2015

nonn

Here sigma is A000203, the sum-of-divisors function.
a(n) is the sum of the n-th row in A085790.
We can divide the set of natural numbers into three classes based on whether a(n)n. The last class is A258914. Are there any n in the second category, i.e., n such that a(n)=n, other than n=1 (see link)?
It is natural to further divide the class a(n)A007369 (not in image of sigma), which is all n for which A054973(n)=0. The second one of these, the case 01) all of A007370 (just one pre-image of n under sigma, equivalently A054973(n)=1), but also includes some terms that have more than one pre-image, see A258931.
If there exists a number n>1 such that a(n)=n, then n > 2.5*10^10. - Giovanni Resta, Jun 15 2015
Row sums of A299762. - Omar E. Pol, Mar 14 2018

			To find a(24), note that the only values of k with sigma(k)=24 are k=14,15,23; therefore a(24)=14+15+23=52.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Mathematics Stack Exchange, A number n which is the sum of all numbers k with sigma(k)=n?

Cf. A000203, A007369, A007370, A054973, A085790, A258914, A258931, A299762.

a[n_] := Sum[k*Boole[DivisorSigma[1, k] == n], {k, 1, n}]; Array[a, 80] (* Jean-François Alcover, Jun 15 2015 *)

PARI
```
a(n)=sum(k=1,n,if(sigma(k)==n,k))
```

first(n)=my(v=vector(n),s); for(k=1,n,s=sigma(k);if(s<=n,v[s]+=k));v \\ Charles R Greathouse IV, Jun 15 2015

a(n) = vecsum(invsigma(n)); \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

A299763 a(n) = 1 + A182986(n).

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

Omar E. Pol, Mar 14 2018

Are these the indices of the rows of A299762 where there is a record?

G. C. Greubel, Table of n, a(n) for n = 1..10000

First differences are in A054541.
Cf. A008864, A028815, A182986, A299762.
Essentially the same as A008864, A028815, A055670, A135731, A175216.

A299763:= func< n | n eq 1 select 1 else NthPrime(n-1) + 1 >;
[A299763(n): n in [1..70]]; // G. C. Greubel, Aug 05 2024

A299763[n_]:= If[n==1, 1, Prime[n-1] +1];
Table[A299763[n], {n,70}] (* G. C. Greubel, Aug 05 2024 *)

def A299763(n): return 1 if n==1 else nth_prime(n-1) + 1
[A299763(n) for n in range(1,71)] # G. C. Greubel, Aug 05 2024

a(n) = A028815(n-1) - [n=1].

a(n) = A008864(n-1) for n >= 2, with a(1) = 1.

A341879 a(n) is the largest d(k) such that sigma(k) = n, where d is the number of divisor function and sigma is the sum of divisors function.

Original entry on oeis.org

1, 0, 2, 2, 0, 2, 3, 2, 0, 0, 0, 4, 3, 2, 4, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 6, 0, 2, 5, 4, 0, 0, 0, 4, 0, 2, 6, 4, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 6, 3, 0, 0, 8, 0, 2, 6, 0, 0, 0, 0, 2, 0, 0, 0, 8, 0, 2, 0, 0, 0, 6, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 8, 9, 0, 6, 0, 0, 8, 0, 6, 0, 0, 0, 2, 0, 6, 0

Offset: 1

Seiichi Manyama, Feb 22 2021

			k that satisfies sigma(k) = 12 is 6 or 11. d(6) = 4 and d(11) = 2. So a(12) = 4.

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

Cf. A000005, A000203, A054973, A057637, A085790, A299762.

a[n_] := Module[{dmax = 0}, Do[If[DivisorSigma[1, k] == n && (d = DivisorSigma[0, k]) > dmax, dmax = d], {k, 1, n}]; dmax]; Array[a, 100] (* Amiram Eldar, Apr 28 2021 *)

a(n) = my(m=0); for(k=1, n, if(sigma(k)==n, m=max(m, numdiv(k)))); m;

cp's OEIS Frontend

A054973 Number of numbers whose divisors sum to n.

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A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

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A085790 Integers sorted by the sum of their divisors.

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A057637 a(n) is the largest number k such that sigma(k) = n, where sigma is the sum of divisors function A000203, or 0 if no such k exists.

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A258913 a(n) is the sum of all numbers k for which sigma(k) = n.

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A299763 a(n) = 1 + A182986(n).

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A341879 a(n) is the largest d(k) such that sigma(k) = n, where d is the number of divisor function and sigma is the sum of divisors function.

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