A054973 -id:A054973 - cp's OEIS frontend

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

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

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

A206036 Numbers m such that sigma(m) = sigma(k) has solution for distinct numbers m and k.

Original entry on oeis.org

6, 10, 11, 14, 15, 16, 17, 20, 21, 23, 24, 25, 26, 28, 30, 31, 33, 34, 35, 38, 39, 40, 41, 42, 44, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

			6 and 11 are in the sequence because sigma(6) = sigma(11) = 12.
7 is not on the list because sigma(7) = 8 and there is no other integer for which sigma(n) = 8.

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

Complement of A211656.

Cf. A000203, A054973, A066074, A158913.

max = 9000; sigmaList = Table[DivisorSigma[1, n], {n, Prime[PrimePi[max]]}]; Select[Range[Floor[Sqrt[max]]], Count[sigmaList, sigmaList[[#]]] > 1 &] (* Alonso del Arte, Feb 06 2012 *)

is(k) = invsigmaNum(sigma(k)) > 1; \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

A275700 a(n) = Product_{d|n} prime(d).

Original entry on oeis.org

2, 6, 10, 42, 22, 390, 34, 798, 230, 1914, 62, 101010, 82, 4386, 5170, 42294, 118, 547170, 134, 951258, 12410, 14694, 166, 170807910, 2134, 24846, 23690, 3285114, 218, 660741510, 254, 5540514, 42470, 49206, 55726, 21399271530, 314, 65526, 68470, 3126785046, 358

Offset: 1

Jaroslav Krizek, Aug 05 2016

nonn

a(n) mod n = 0 for n: 1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, ...

			a(4) = 42 because the divisors of 4 are: 1, 2 and 4; and prime(1) * prime(2) * prime(4) = 2 * 3 * 7 = 42.

Cf. A007445 (Sum_{d|n} prime(d)).
A version for binary indices is A034729.
Partitions of this type are counted by A054973, strict case of A371284.
The sorted version is A371283, squarefree case of A371288.
These numbers have products A371286, unsorted version A371285.
A000005 counts divisors, row-lengths of A027750.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A001221, A005179, A007416.

[(&*[NthPrime(d): d in Divisors(n)]): n in [1..100]]

Table[Times@@(Prime[#]&/@Divisors[n]),{n,50}] (* Harvey P. Dale, Jun 16 2017 *)

a(n) = my(d=divisors(n)); prod(i=1, #d, prime(d[i])) \\ Felix Fröhlich, Aug 05 2016

use ntheory ":all"; sub a275700 { vecprod(map { nth_prime($) } divisors($[0])); } # Dana Jacobsen, Aug 09 2016

A002192 Least integer with A000203(a(n)) = A002191(n), where A002191 = range of the sum-of-divisors function A000203.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 9, 13, 8, 10, 19, 14, 12, 29, 16, 21, 22, 37, 18, 27, 20, 43, 33, 34, 28, 49, 24, 61, 32, 67, 30, 73, 45, 57, 44, 40, 36, 50, 42, 52, 101, 63, 85, 109, 91, 74, 54

Offset: 1

N. J. A. Sloane

This is the least integer with the increasing sigma value A002191(n). For integers sorted on the ordered sigma values A007609(n), see A085790. - Lekraj Beedassy, Oct 08 2004

The sigma function (A000203) can't have a left nor a right inverse since it is neither injective nor surjective. The first column of the table A085790 (undefined when the row length A054973(n) = 0 <=> no x has sigma(x) = n) or A051444 (which has zeros filled in for these undefined values) are right-inverse of sigma on A002191 = range of sigma: one has A000203(A051444(n)) = A000203(A085790(n,1)) = n for all n in A002191 <=> A054973(n) > 0 <=> row A085790(n,.) nonempty <=> there is x with sigma(x) = n. Since sigma(6) = sigma(11) = 12, a hypothetical left inverse g must satisfy g(12) = 6 and g(12) = 11 which is impossible. Restricted to this list A002192 of smallest indices for the possible values of sigma, there exists a left inverse g such that g(sigma(x)) = x for all x in A002192. This equation defines the function g, i.e., g(A002191(n)) := a(n). A different left inverse exists on the set of largest pre-images for the possible values of sigma, {A085790(n,A054973(n)); n in A002191} = {1, 2, 3, 5, 4, 7, 11, 9, 13, 8, 17, 19, 23, 12, 29, 25, 31, 22, 37, 18, 27, 41, 43, ...}. - M. F. Hasler, Nov 21 2019

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

A051444 is a better version of this sequence.

Cf. A000203, A002191, A007609, A007626, A054973, A085790.

m = 1000; Clear[f]; f[k_] := f[k] = Split[{DivisorSigma[1, #], #}& /@ Range[3k] // Sort, #1[[1]] == #2[[1]]&][[1 ;; m, 1]][[All, 2]]; f[k = m]; f[k = k+m]; While[f[k] != f[k, m], k = k+m]; A002192 = f[k] (* Jean-François Alcover, Oct 15 2015 *)

A063974 Number of terms in inverse set of usigma = sum of unitary divisors = A034448.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 6, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0

Offset: 1

Labos Elemer, Sep 05 2001

nonn

			usigma(x) = 288, invusigma(288) = {138,154,165,168,213,235,248,253}, so a(288) = 8, the number of all terms in the inverse set.

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

Cf. A034444, A034448, A051444, A054973, A057637, A308041.

Size of set {x; usigma(x) = n}.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308041. - Amiram Eldar, Dec 23 2024

A074753 Number of integers k such that sigma(k) < n.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 6, 6, 8, 9, 10, 11, 11, 11, 13, 13, 14, 14, 14, 14, 17, 17, 17, 17, 18, 18, 19, 21, 23, 23, 23, 23, 24, 24, 25, 26, 27, 27, 30, 30, 31, 31, 31, 31, 34, 34, 34, 34, 34, 34, 36, 36, 38, 39, 39, 39, 42, 42, 43, 44, 44, 44, 44, 44, 45, 45, 45, 45, 50, 50

Offset: 1

Benoit Cloitre, Sep 28 2002

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert E. Dressler, An elementary proof of a theorem of Erdős on the sum of divisors function, Journal of Number Theory, Vol. 4, No. 6 (1972), pp. 532-536.

Partial sums of A054973.

Cf. A000203, A308039.

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
for n from 1 to N-2 do
  s:= numtheory:-sigma(n)+1;
  if s <= N then V[s]:= V[s]+1 fi;
od:
ListTools:-PartialSums(V); # Robert Israel, Jan 08 2018

Table[Length[Select[Range[n], DivisorSigma[1,#] < n&]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *)

a(n)=sum(i=1,n,if(1+sign(sigma(i)-n),0,1))

list(nmax) = my(s = 0); for(n = 1, nmax, s += invsigmaNum(n); print1(s, ", ")); \\ Amiram Eldar, Dec 23 2024, using Max Alekseyev's invphi.gp

a(n) = card( k : sigma(k) < n ).
a(n) is asymptotic to c*n with c = 0.67...
a(n) = c * n + o(n), where c = 0.6727383... = A308039 (Dressler, 1972). - Amiram Eldar, Dec 23 2024

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

A175192 a(n) = characteristic function of numbers k such that A000203(m) = k has solution, where A000203(m) = sums of divisors of m.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Jaroslav Krizek, Mar 01 2010

nonn

a(n) = 1 if A000203(m) = n for some m, else 0.

a(n) = 1 for n such that A054973(n) >= 1. a(n) = 0 for n such that A054973(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Characteristic function of A002191.

Cf. A000203, A054973.

nn = 200; t = Union[Select[DivisorSigma[1, Range[nn]], # <= nn &]]; t2 = Table[0, {200}]; t2[[t]] = 1; t2 (* T. D. Noe, Jan 24 2012 *)

up_to = 65537
v175192 = vector(up_to);
for(k=1, up_to, t=sigma(k); if(t<=up_to, v175192[t] = 1)); \\ See also code in A054973.
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,v175192,"b175192_upto65537.txt");
\\ Antti Karttunen, Oct 20 2017

More terms from Antti Karttunen, Oct 20 2017

A216200 Number of disjoint trees that appear while iterating the sum of divisors function up to n.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 6, 5, 5, 5, 5, 6, 7, 6, 7, 7, 8, 9, 10, 8, 9, 10, 11, 11, 12, 12, 11, 10, 11, 12, 13, 13, 14, 14, 14, 14, 15, 13, 14, 14, 15, 16, 17, 15, 16, 17, 18, 19, 20, 19, 20, 19, 19, 20, 21, 19, 20, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27

Offset: 1

Michel Marcus, Mar 12 2013

nonn

A tree like (2, 3, 4, 7, 8) contains all numbers below 8 such that iterating the sum of divisors function to any of them, while staying below 8, will lead to 8.

Inspired by the article in link, where a (p1, p2, p3)-tree is defined with p1 the smallest number in the tree, and p2, p3, such that all sequences {sigma^i(n)} (iterations of sigma), with p1 <= n <= p2 and sigma^i(n) < p3 have nonempty intersection with {sigma^i(p1)}. For instance, 21 (p1, 200, 10^10)-trees and 64 (p1, 1000, 10^100)-trees were found.

			For n=23, there are 10 disjoint trees: (1), (2, 3, 4, 7, 8, 15), (5, 6, 11, 12), (9, 13, 14), (10, 17, 18), (16), (19, 20), (21), (22), (23). With the arrival of 24, 3 trees are united, that is those that contain 15, 14 and 23, so that there are now 8 trees.
Some further values: a(100) = 33, a(500) = 167, a(1000) = 333.
Further values: a(10^4) = 3282, a(10^5) = 32739, a(10^6) = 327165, a(10^7) = 3272134. - _M. F. Hasler_, Nov 19 2019

M. F. Hasler, Table of n, a(n) for n = 1..10000
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.

Cf. A000203.

Cf. A257669, A257670: size and smallest number of subtree rooted at n.

A216200_vec(N)={my(C=Map(), s, c); vector(N, n, mapisdefined(C,n)&& c+=mapget(C,n) + mapdelete(C,n); mapput(C, s=sigma(n), if(mapisdefined(C,s), mapget(C,s))+1); n-c)} \\ Use allocatemem() for N >= 10^6.
A216200(n)={my(C=Map(), s); n-sum(n=2,n, mapput(C, s=sigma(n), if(mapisdefined(C,s), mapget(C,s))+1); if(mapisdefined(C,n), mapget(C,n) + mapdelete(C,n)))} \\ (slightly faster to compute a single value)
tree(n)=[n,if(n>1,apply(self,invsigma(n)),"fixed point")] \\ to create the tree rooted in n. (End)

For n > 1, a(n) = a(n-1) + 1 - A054973(n), a(1) = 1. - Michel Marcus, Oct 22 2013

It appears that a(n)/n = 0.32721... + O(1/sqrt(n)). - M. F. Hasler, Nov 19 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A206036 Numbers m such that sigma(m) = sigma(k) has solution for distinct numbers m and k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A275700 a(n) = Product_{d|n} prime(d).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Perl

A002192 Least integer with A000203(a(n)) = A002191(n), where A002191 = range of the sum-of-divisors function A000203.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A063974 Number of terms in inverse set of usigma = sum of unitary divisors = A034448.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A074753 Number of integers k such that sigma(k) < n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords