A238895 -id:A238895 - cp's OEIS frontend

A238896 Number of times that the number A238895(n) occurs in the sum of proper divisors function (A001065).

0, 1, 2, 3, 5, 6, 8, 9, 10, 13, 15, 20, 21, 24, 25, 27, 32, 34, 41, 44, 46, 56, 59, 70, 74, 76, 86, 92, 99, 115, 116, 129, 138, 145, 156, 164, 169, 171, 193, 199, 222, 223, 242, 246, 268, 270, 278, 295, 304, 334, 343, 365, 397, 398, 433, 452, 471, 482, 521

Offset: 1

T. D. Noe, Mar 10 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..245

Cf. A001065, A005114, A048138, A238895.

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

a(n) = A048138(A238895(n)). - Amiram Eldar, Sep 23 2022

A239625 Irregular triangle of numbers k that have A238895(n) as the sum of the proper divisors of each k.

Original entry on oeis.org

4, 6, 25, 18, 51, 91, 32, 125, 161, 209, 221, 75, 215, 287, 407, 527, 551, 98, 175, 335, 671, 767, 1007, 1247, 1271, 581, 869, 1241, 1349, 1541, 1769, 1829, 1961, 2021, 545, 749, 1133, 1313, 1649, 2573, 2993, 3053, 3149, 3233, 243, 791, 1199, 1391, 1751, 1919

Offset: 2

T. D. Noe, Mar 28 2014

It is interesting that these numbers have a very simple factorization. For example, the terms in row 6 factor as 3*5^2, 5*43, 7*41, 11*37, 17*31, and 19*29.

			The rows are
1: {empty}
2: 4,
3: 6, 25,
4: 18, 51, 91,
5: 32, 125, 161, 209, 221,
6: 75, 215, 287, 407, 527, 551,
7: 98, 175, 335, 671, 767, 1007, 1247, 1271,
8: 581, 869, 1241, 1349, 1541, 1769, 1829, 1961, 2021

T. D. Noe, Rows n = 2..51 of irregular triangle, flattened

Cf. A238895.

nn = 200; s = Table[0, {nn}]; d = Table[DivisorSigma[1, n] - n, {n, (nn/2)^2}]; Do[If[0 < d[[n]] <= nn, s[[d[[n]]]]++], {n, (nn/2)^2}]; t = {}; mx = -1; Do[If[s[[n]] > mx, mx = s[[n]]; AppendTo[t, {n, mx}]], {n, 2, nn}]; t2 = Transpose[t][[1]]; Table[Flatten[Position[d, n]], {n, t2}]

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

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

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

A325177 Unitary highly touchable numbers: Numbers m > 1 such that a record number of numbers k have m as the sum of the proper unitary divisors of k.

Original entry on oeis.org

2, 6, 8, 12, 22, 33, 49, 55, 67, 79, 91, 115, 121, 151, 169, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1171, 1231, 1261, 1321, 1471, 1561, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251

Offset: 1

Amiram Eldar, Sep 05 2019

nonn

The unitary version of A238895.

Amiram Eldar, Table of n, a(n) for n = 1..78
Amiram Eldar, Table of n, a(n), A324938(a(n)) for n = 1..78

Cf. A034448, A034460, A063948, A238895, A324938.

us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  m = 300; v = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A331974 Infinitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper infinitary divisors of k.

Original entry on oeis.org

2, 6, 8, 17, 21, 37, 49, 55, 67, 79, 85, 91, 121, 151, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1231, 1261, 1471, 1561, 1651, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251, 5461, 6091

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

The infinitary version of A238895.

			a(1) = 2 since it is the first number which is not the sum of proper infinitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper infinitary divisors of one number: 6 = A126168(6).
a(3) = 8 since it is the least number which is the sum of proper infinitary divisors of 2 numbers: 8 = A126168(10) = A126168(12).

Amiram Eldar, Table of n, a(n) for n = 1..76 (terms below 30000)

Cf. A049417, A077609, A126168, A238895, A324277, A325177, A331972, A331973.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A064440 Numbers k such that A048138(k) is a prime and sets a new record for such primes.

Original entry on oeis.org

6, 21, 31, 79, 121, 265, 271, 379, 541, 631, 1051, 1351, 1597, 1711, 2071, 2401, 2551, 2761, 2881, 2941, 3811, 4111, 4471, 4531, 4621, 5251, 6091, 8341, 8371, 8401, 8821, 9871, 11551, 13651, 16171, 19531, 21211, 22681, 23101, 27931, 30661, 32551, 33811, 37591

Offset: 1

Jason Earls, Oct 01 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..188

Cf. A001065, A048138, A238895, A238896.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i, v = {}, m = 0}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]]++], {n, 2, (max - 1)^2}]; Do[If[s[[i]] > m && PrimeQ[s[[i]]], m = s[[i]]; AppendTo[v, i]], {i, 1, max}]; v]; seq[2000] (* Amiram Eldar, Sep 23 2022 *)

f(n) = my(s=0); for(x=1,n^2, if(sigma(x)-x==n,s++)); s;
a=0; for(n=1,10^5,x=f(n); if(isprime(x),b=x; if(b>a,a=b; print1(n, ", "))))

Corrected and extended by Matthew Conroy, Oct 02 2001

a(11)-a(44) from Amiram Eldar, Oct 01 2019

A331972 Bi-unitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper bi-unitary divisors of k.

Original entry on oeis.org

2, 6, 8, 17, 29, 31, 55, 79, 91, 115, 121, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1231, 1261, 1471, 1561, 1651, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251, 5461, 6091, 6511, 6931

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

The corresponding record values are 0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 14, 15, ...

The bi-unitary version of A238895.

			a(1) = 2 since it is the first number which is not the sum of proper bi-unitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper bi-unitary divisors of one number: 6 = A331970(6).
a(3) = 8 since it is the least number which is the sum of proper bi-unitary divisors of 2 numbers: 8 = A331970(10) = A331970(12).

Amiram Eldar, Table of n, a(n) for n = 1..73 (terms below 30000)

Cf. A188999, A238895, A324276, A325177, A331970, A331974.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A356445 a(n) is the number of times that A064440(n) occurs as the sum of proper divisors function (A001065).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 59, 61, 67, 79, 83, 97, 101, 109, 113, 127, 131, 139, 149, 151, 193, 199, 223, 227, 229, 277, 283, 317, 397, 433, 521, 541, 577, 607, 677, 743, 757, 811, 863, 881, 911, 971, 1031, 1049, 1063, 1093, 1249, 1319, 1373, 1433, 1489

Offset: 1

Amiram Eldar, Sep 23 2022

nonn

The record values correspond to the terms of A064440.

All the terms are primes by definition.

Amiram Eldar, Table of n, a(n) for n = 1..188

Cf. A001065, A048138, A064440, A238895, A238896.

seq[max_] := Module[{s = Table[0, {n, 1, max}], i, v = {}, m = 0}, Do[If[(i = DivisorSigma[1, n] - n) <= max, s[[i]]++], {n, 2, (max - 1)^2}]; Do[If[s[[i]] > m && PrimeQ[s[[i]]], m = s[[i]]; AppendTo[v, s[[i]]]], {i, 1, max}]; v]; seq[2000]

a(n) = A048138(A064440(n)).

cp's OEIS Frontend

A238896 Number of times that the number A238895(n) occurs in the sum of proper divisors function (A001065).

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A239625 Irregular triangle of numbers k that have A238895(n) as the sum of the proper divisors of each k.

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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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Formula

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

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Maple

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Extensions

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

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Mathematica

PARI

Extensions

A325177 Unitary highly touchable numbers: Numbers m > 1 such that a record number of numbers k have m as the sum of the proper unitary divisors of k.

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Mathematica

A331974 Infinitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper infinitary divisors of k.

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