A005114 -id:A005114 - cp's OEIS frontend

A177900 Smallest positive integer k such that the sum of the proper divisors of k gives the n-th prime, or 0 if prime(n) belongs to A005114.

0, 4, 0, 8, 21, 27, 39, 65, 57, 115, 32, 155, 63, 50, 129, 235, 265, 371, 305, 201, 98, 365, 237, 171, 245, 291, 485, 309, 325, 327, 128, 189, 279, 917, 1507, 1529, 242, 785, 489, 835, 865, 1211, 385, 605, 579, 965, 338, 2321, 669, 1115, 687, 1165, 399, 2629, 549, 1285, 595, 1841, 1025, 603, 1385, 715, 3809, 921, 1175, 939, 4121, 1655, 805, 1325

Offset: 1

John W. Layman, Dec 15 2010

nonn

Suggested by the following comment of T. D. Noe (see A037020): "Assuming the Goldbach conjecture, it is easy to show that all primes, except 2 and 5, are the sum of the proper divisors of some number."

			27 and 35 have the proper divisors {1,3,9} and {1,5,7}, respectively, both of which sum to prime(6)=13, and no positive integer smaller than 27 has this property.  Thus a(6)=27.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A037020, A005114.

N:= 1000: # to get the first N terms
A:= 'A': A[1]:= 0: A[3]:= 0:
Primes:= [seq(ithprime(i),i=1..N)]:
count:= 0:
for k from 1  while count < N-2 do
  m:= numtheory:-sigma(k)-k;
  if member(m,Primes,'n') and not assigned(A[n]) then
     count:= count+1;
     A[n]:= k;
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Jul 09 2014

f[n_] := DivisorSigma[1, n] - n; k = 1; t = Table[0, {1000}]; While[ k < 24001, a = f@ k; If[a < 1001 && t[[a]] == 0, t[[a]] = k]; k++]; tt = Table[t[[n]], {n, Prime@ Range@ 168}] (* Robert G. Wilson v, Jul 09 2014 *)

a(n) = {p = prime(n); if ((p != 2) && (p != 5), k = 2; while ((sigma(k) - k) != p, k++); k;);} \\ Michel Marcus, Jul 09 2014

a(n) = A070015(p_n) where p runs through the primes. - Robert G. Wilson v, Jul 09 2014

Offset corrected to 1 by Michel Marcus, Jul 09 2014

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106

Offset: 1

N. J. A. Sloane, R. K. Guy

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
Fixed points are in A000396. - Alois P. Heinz, Mar 10 2024

			x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - _Omar E. Pol_, Nov 24 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 91.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [alternative scanned copy].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences

Least inverse: A070015, A359132.
Values taken: A078923, values not taken: A005114.
Records: A034090, A034091.
First differences: A053246, partial sums: A153485.
a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013
Row sums of A141846 and of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik, May 20 2012
Alternating row sums of A231347. - Omar E. Pol, Jan 02 2014
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
For n > 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).

a001065 n = a000203 n - n  -- Reinhard Zumkeller, Sep 15 2011

[SumOfDivisors(n)-n: n in [1..100]]; // Vincenzo Librandi, May 06 2015

A001065 := proc(n)
    numtheory[sigma](n)-n ;
end proc:
seq( A001065(n),n=1..100) ;

Table[ Plus @@ Select[ Divisors[ n ], #Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)

numlib::sigma(n)-n$ n=1..81 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */

from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)-n # Chai Wah Wu, Nov 04 2022

[sigma(n, 1)-n for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005
a(n) = A155085(-n). - Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - José de Jesús Camacho Medina, Feb 23 2021

A045572 Numbers that are odd but not divisible by 5.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153

Offset: 1

Jeff Burch, Dec 11 1999

Contains the repunits R_n, (A000042 or A002275): For any m in the sequence (divisible by neither 2 nor 5), Euler's theorem (i.e., m | 10^m - 1 = 9*R_n) guarantees that R_n is always some multiple of m (see A099679) and thus forms a subsequence. - Lekraj Beedassy, Oct 26 2004
Inverse formula: n = 4*floor(a(n)/10) + floor((a(n) mod 10)/3) + 1. - Carl R. White, Feb 06 2008
Numbers ending with 1, 3, 7 or 9. - Lekraj Beedassy, Apr 04 2009
Complement of A065502. - Reinhard Zumkeller, Nov 15 2009
Union of evenish and oddish numbers, cf. A045797, A045798. - Reinhard Zumkeller, Dec 10 2011
Numbers k such that k^(4*j) mod 10 = 1, for any j. - Gary Detlefs, Jan 03 2012
Numbers coprime to 10. - Charles R Greathouse IV, Sep 05 2013
This is also the sequence of numbers such that all their divisors are the sum of the proper divisors of some number (see A001065 (sum of proper divisors) and A078923 (possible values of sigma(n)-n)). This is due to the fact that in the set of untouchable numbers (A005114) there are only 2 prime numbers (2 and 5) and all other terms are even composite. - Michel Marcus, Jun 14 2014
Numbers n for which A001589(n) is divisible by 5. - Bruno Berselli, Jun 18 2014
For a(n) > 1, positive integers x such that the decimal representation of 1/x is purely periodic after the decimal point (1/x is a repeating decimal with no non-repeating portion). - Doug Bell, Aug 05 2015
The asymptotic density of this sequence is 2/5. - Amiram Eldar, Oct 18 2020

			a(18) = 10*floor(17/4) + 2*floor( (4*(17 mod 4) + 1)/3 ) + 1
      = 10*4 + 2*floor( (4*(1)+1)/3 ) + 1
      = 40 + 2*floor(5/3) + 1
      = 40 + 2*1 + 1
      = 43.
G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Relative complement of A017329 in A005408.

Cf. A000035, A000042, A001065, A001589, A002275, A005114, A045797, A045798, A065502, A078923, A079998, A082768 (numbers that begin with 1, 3, 7 or 9), A085820, A099679.

a045572 n = a045572_list !! (n-1)
a045572_list = filter ((/= 0) . (`mod` 5)) a005408_list
-- Reinhard Zumkeller, Dec 10 2011

[ 2*n + 2*Floor((n-3)/4) + 1: n in [1..70] ]; // Vincenzo Librandi, Aug 01 2011

A045572:=n->2*n + 2*floor((n-3)/4) + 1; seq(A045572(n), n=1..50); # Wesley Ivan Hurt, Jun 14 2014

Flatten[Table[10n + {1, 3, 7, 9}, {n, 0, 19}]] (* Alonso del Arte, Jan 13 2012 *)
Select[2Range@78 -1, Mod[#, 5] > 0 &] (* Robert G. Wilson v, Apr 02 2017 *)
Map[(1/2*(5*# + Mod[3*# + 2, 4] - 4))&, Range[10^3]] (* Mikk Heidemaa, Nov 23 2017 *)

a(n)=10*(n>>2)+[-1,1,3,7][n%4+1] \\ Charles R Greathouse IV, Jul 31 2011

is(n)=gcd(n,10)==1 \\ Charles R Greathouse IV, Sep 05 2013

{a(n) = 2*n - 1 + (n+1) \ 4 * 2}; /* Michael Somos, Jun 15 2014 */

def A045572(n): return 2*(n+(n+1)//4) - 1 # Chai Wah Wu, Jan 08 2022

a(n) = 10*floor((n-1)/4) + 2*floor( (4*((n-1) mod 4) + 1)/3 ) + 1; a(n) = a(n-1) + 2 + 2*floor(((x+6) mod 10)/9). - Carl R. White, Feb 06 2008
a(n) = 2*n + 2*floor((n-3)/4) + 1. - Kenneth Hammond (weregoose(AT)gmail.com), Mar 07 2008
a(n) = -1 + 2*n + 2*floor((n+1)/4). - Kenneth Hammond (weregoose(AT)gmail.com), Mar 25 2008
From R. J. Mathar, Sep 22 2009: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1+x) * (x^2+1) * (x-1)^2). (End)
A000035(a(n))*(1 - A079998(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2009
a(n) = (10*n + 2*(-1)^(n*(n+1)/2) - (-1)^n - 5)/4. - Bruno Berselli, Nov 06 2011
G.f.: x * (1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x) * (1 - x^4)). - Michael Somos, Jun 15 2014
a(1 - n) = -a(n) for all n in Z. - Michael Somos, Jun 15 2014
0 = (a(n) - 2*a(n+1) + a(n+2)) * (a(n) - 4*a(n+2) + 3*a(n+3)) for all n in Z. - Michael Somos, Jun 15 2014
From Mikk Heidemaa, Nov 22 2017: (Start)
a(n) = (1/2)*(5*n + ((3*n + 2) mod 4) - 4);
a(n) = (1/4)*((-1)^(n + 1) + 10*n + 2*cos((n*Pi)/2) - 2*sin((n*Pi)/2) - 5);
a(n) = (1/4)*((-1)^(1 + n) + (1 - i)*exp(-(1/2)*i*n*Pi) + (1 + i)*exp(i*n*Pi/2) + 10*n - 5) (for n > 0), where i is the imaginary unit. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(10-2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 12 2021
E.g.f.: (2 + cos(x) + (5*x - 3)*cosh(x) - sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Dec 07 2022

A007369 Numbers n such that sigma(x) = n has no solution.

Original entry on oeis.org

2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

With an initial 1, may be constructed inductively in stages from the list L = {1,2,3,....} by the following sieve procedure. Stage 1. Add 1 as the first term of the sequence a(n) and strike off 1 from L. Stage n+1. Add the first (i.e. leftmost) term k of L as a new term of the sequence a(n) and strike off k, sigma(k), sigma(sigma(k)),.... from L. - Joseph L. Pe, May 08 2002
This sieve is a special case of a more general sieve. Let D be a subset of N and let f be an injection on D satisfying f(n) > n. Define the sieve process as follows: 1. Start with the empty sequence S and let E = D. 2. Append the smallest element s of E to S. 3. Remove s, f(s), f(f(s)), f(f(f(s))), ... from E. 4. Go to step 2. After this sieving process, S = D - f(D). To get the current sequence, take f = sigma and D = {n | n >= 2}. - Max Alekseyev, Aug 08 2005
By analogy with the untouchable numbers (A005114), these numbers could be named "sigma-untouchable". - Daniel Lignon, Mar 28 2014
The asymptotic density of this sequence is 1 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			a(4) = 10 because there is no x < 10 whose sigma(x) = 10.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Complement of A002191.
See A083532 for the gaps, i.e., first differences.
See A048995 for the missed sums of nontrivial divisors.

a = {}; Do[s = DivisorSigma[1, n]; a = Append[a, s], {n, 1, 115} ]; Complement[ Table[ n, {n, 1, 115} ], Union[a] ]

list(lim)=my(v=List(),u=vectorsmall(lim\1),t); for(n=1,lim, t=sigma(n); if(t<=lim, u[t]=1)); for(n=2,lim, if(u[n]==0, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Mar 09 2017

A007369_list(LIM,m=0,L=List(),s)={for(n=2,LIM,(s=sigma(n-1))>LIM || bittest(m,s) || m+=1<M. F. Hasler, Mar 12 2018

A175192(a(n)) = 0, A054973(a(n)) = 0. - Jaroslav Krizek, Mar 01 2010

a(n) < 2n + sqrt(8n). - Charles R Greathouse IV, Oct 23 2015

More terms from David W. Wilson

A054024 Sum of the divisors of n reduced modulo n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 8, 1, 4, 1, 10, 9, 15, 1, 3, 1, 2, 11, 14, 1, 12, 6, 16, 13, 0, 1, 12, 1, 31, 15, 20, 13, 19, 1, 22, 17, 10, 1, 12, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 12, 17, 8, 23, 32, 1, 48, 1, 34, 41, 63, 19, 12, 1, 58, 27, 4, 1, 51, 1, 40, 49, 64, 19, 12, 1, 26, 40

Offset: 1

Asher Auel, Jan 19 2000

If a(n) = 0, then n is a multiply-perfect number (A007691). - Alonso del Arte, Mar 30 2014

			a(12) = 4 because sigma(12) = 28 and 28 == 4 (mod 12).
a(13) = 1 because 13 is prime.
a(14) = 10 because sigma(14) = 24 and 24 == 10 (mod 14).

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe, terms 1001..20000 from Alois P. Heinz).
Walter Nissen, Abundancy : Some Resources.

Cf. A000203 (sigma), A005114 (untouchable numbers), A007691 (positions of 0's), A045768, A045769, A088834, A045770, A076496, A159907.

a054024 n = mod (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

with(numtheory): seq(sigma(i) mod i, i=1..100);

Table[Mod[DivisorSigma[1, n], n], {n, 80}] (* Alonso del Arte, Mar 30 2014 *)

a(n)=sigma(n)%n \\ Charles R Greathouse IV, Nov 04 2014

a(n) = sigma(n) mod n.

a(p) = 1 for p prime.

A048138 a(n) = number of m such that sum of proper divisors of m (A001065(m)) is n.

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 2, 1, 5, 2, 3, 1, 3, 1, 4, 1, 1, 3, 4, 2, 5, 2, 3, 2, 3, 1, 6, 2, 4, 0, 3, 2, 6, 1, 5, 1, 3, 1, 6, 2, 3, 3, 6, 1, 6, 1, 2, 1, 5, 1, 8, 3, 4, 3, 5, 1, 7, 1, 6, 1, 4, 1, 8, 1, 5, 0, 5, 2, 9, 2, 4, 1, 4, 0, 9, 1, 3, 2, 6, 1, 8, 2, 7, 4

Offset: 2

Naohiro Nomoto

The offset is 2 since there are infinitely many numbers (all the primes) for which A001065 = 1.
The graph of this sequence, shifted by 1, looks similar to that of A061358, which counts Goldbach partitions of n. - T. D. Noe, Dec 05 2008
For n > 2, a(n) <= A000009(n) as all divisor lists must have distinct values. - Roderick MacPhee, Sep 13 2016
The smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k is A125601(n). - Bernard Schott, Mar 23 2023

			a(6) = 2 since 6 is the sum of the proper divisors of 6 and 25.

T. D. Noe, Table of n, a(n) for n = 2..10000
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.

Cf. A001065, A064440, A125601, A238895, A238896 (records).

Cf. A005114, A057709, A057710, A160133.

with(numtheory): for n from 2 to 150 do count := 0: for m from 1 to n^2 do if sigma(m) - m = n then count := count+1 fi: od: printf(`%d,`,count): od:

list(n)=my(v=vector(n-1),k); for(m=4,n^2, k=sigma(m)-m; if(k>1 & k<=n, v[k-1]++)); v \\ Charles R Greathouse IV, Apr 21 2011

From Bernard Schott, Mar 23 2023: (Start)
a(n) = 0 iff n is in A005114 (untouchable numbers).
a(n) = 1 iff n is in A057709 ("hermit" numbers).
a(n) = 2 iff n is in A057710.
a(n) > 1 iff n is in A160133. (End)

More terms from James Sellers, Feb 19 2001

A152454 Irregular triangle in which row n lists the numbers whose proper divisors sum to n.

Original entry on oeis.org

4, 9, 6, 25, 8, 10, 49, 15, 14, 21, 121, 27, 35, 22, 169, 16, 33, 12, 26, 39, 55, 289, 65, 77, 34, 361, 18, 51, 91, 20, 38, 57, 85, 529, 95, 119, 143, 46, 69, 133, 28, 115, 187, 841, 32, 125, 161, 209, 221, 58, 961, 45, 87, 247, 62, 93, 145, 253, 24, 155, 203, 299, 323, 1369

Offset: 2

T. D. Noe, Dec 05 2008

In an aliquot sequence, all numbers in row n can be predecessors of n. This sequence is a permutation of the composite numbers; number k appears in row A001065(k). We start with n=2 because every prime would be in row 1. Note that row 2 is empty -- as are all the rows listed in A005114. Row n contains A048138(n) numbers. When n is prime, the largest number in row n+1 is n^2. When n>7 is odd, the largest number in row n is less than ((n-1)/2)^2 and (if a strong form of the Goldbach conjecture is true) has the form pq, with primes p

In row n, the first term is A070015(n), and the last term is A135244(n). - Michel Marcus, Nov 11 2014

The first row with several terms is row(6), where the difference between extreme terms is 25-6=19. The next row with a smaller difference is row(13) with a difference 35-27=8. And the next one is row(454) with a difference 602-596=6. Is there a next row with a smaller difference? - Michel Marcus, Nov 11 2014

			Irregular triangle starts:
  ; (empty row at n=2)
  4;
  9;
  ; (empty row at n=5)
  6, 25;
  8;
  10, 49;
  15;
  14;
  21;
  121;
  27, 35;
  22, 169;
  16, 33;
  12, 26;
  39, 55;
  289;
  ...

T. D. Noe, Rows n=2..1000 of triangle, flattened
Michel Marcus, Rows n=2..1000 of triangle, not flattened

Cf. A001065, A005114, A048138.

Cf. A070015, A135244, A135245.

N:= 100: # for rows 2 to N, flattened
for s from 2 to N do B[s]:= NULL od:
for k from 1 to N^2 do
  if not isprime(k) then
    s:= numtheory:-sigma(k)-k;
    if s <= N then
       B[s]:= B[s],k;
    fi
  fi
od:
seq(B[s],s=2..N); # Robert Israel, Nov 11 2014

nn=100; s=Table[{},{nn}]; Do[k=DivisorSigma[1,n]-n; If[1

row(n) = select(x->(sigma(x)-x)==n, [1..n^2]); \\ Michel Marcus, Feb 25 2025

A238895 Numbers m > 1 such that a record number of numbers k have m as the sum of the proper divisors of k.

Original entry on oeis.org

2, 3, 6, 21, 31, 49, 73, 91, 115, 121, 169, 211, 301, 331, 361, 391, 421, 511, 631, 721, 781, 841, 1051, 1261, 1471, 1561, 1681, 1891, 2101, 2311, 2521, 2731, 3151, 3361, 3571, 3991, 4201, 4411, 4621, 5251, 5461, 6091, 6511, 6721, 6931, 7771, 7981, 8191, 9031

Offset: 1

T. D. Noe, Mar 10 2014

nonn

The number of times that a(n) appears in A001065 is A238896(n).
By analogy with the untouchable numbers (A005114) and the highly composite numbers (A002182), these numbers can be named "highly touchable" (see Lignon). - Daniel Lignon, Mar 21 2014
Indices of record values in A048138. - Franklin T. Adams-Watters, Jul 27 2014

			For 2, there are no numbers.
For 3, there is 1 number: 4.
For 6, there are 2 numbers: 6 and 25.
For 21, there are 3 numbers: 18, 51, 91.
For 31, there are 5 numbers: 32, 125, 161, 209, 221.
For 49, there are 6 numbers: 75, 215, 287, 407, 527, 551.

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Editions Ellipses, 2012, see p. 317 (in French).

Amiram Eldar, Table of n, a(n) for n = 1..245 (terms 1..139 from Daniel Mondot)

Cf. A001065, A005114, A048138, A238896.
Cf. A152454 (row n lists the numbers whose proper divisors sum to n).
Cf. A239625 (irregular table giving the rows of numbers that produce a(n)).

nn = 1000; s = Table[0, {nn}]; Do[k = DivisorSigma[1, n] - n; If[0 < k <= nn, s[[k]]++], {n, nn^2}]; t = {}; mx = -1; Do[If[s[[n]] > mx, mx = s[[n]]; AppendTo[t, {n, mx}]], {n, 2, nn}]; Transpose[t][[1]]

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cp's OEIS Frontend

A231964 Lesser member of untouchable twin pairs, that is, such that a(n) and a(n)+2 are untouchable (A005114).

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A231965 Smallest integer starting a group of exactly n consecutive untouchable numbers (A005114) with term differences of 2.

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A177900 Smallest positive integer k such that the sum of the proper divisors of k gives the n-th prime, or 0 if prime(n) belongs to A005114.

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A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

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A045572 Numbers that are odd but not divisible by 5.

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A007369 Numbers n such that sigma(x) = n has no solution.

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