A051444 -id:A051444 - cp's OEIS frontend

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

1, 2, 0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56

Offset: 0

Labos Elemer, Apr 12 2002

For odd n >= 9, a(n) <= A073046((n-1)/2). - Max Alekseyev, Sep 01 2025

			For n=128: a(128)=16129, divisors={1,127,16129}, 1+127=sigma(n)-n=128 and 16129 is the smallest.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 9884 terms from Richard J. Mathar)
Mersenne Forum, Given sigma(n)-n, find the smallest possible n

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

See A359132 for another version.

f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1000000}]; t

a(n) = min(x, A001065(x)=n) or a(n)=0 if n is an untouchable number (i.e., if from A005114).

a(0)=1 prepended by Max Alekseyev, Sep 01 2025

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

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

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.

A007626 Sum of divisors of superabundant numbers (A004394).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 360, 546, 744, 1170, 2418, 2880, 4368, 5952, 9360, 19344, 39312, 59520, 99944, 112320, 232128, 471744, 714240, 1199328, 1451520, 2437344, 2926080, 3249792, 6604416, 9999360

Offset: 1

Walter Nissen

Local maxima of sigma(n), the sum of divisors function A000203.

Same as A063072 for the first 19 terms. - T. D. Noe, Jul 01 2008

Amiram Eldar, Table of n, a(n) for n = 1..6863 (terms 1..500 from T. D. Noe)

Cf. A002192, A000203, A051444, A004394.

See A034885 and A002093 for another version.

Reap[ For[ n=1; a=0, n <= 3*10^6, n++, s = DivisorSigma[1, n]; b = s/n; If[ b>a, a=b; Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Apr 02 2013 *)
Join[{1},DeleteDuplicates[Select[{#[[1]],#[[2]],#[[2]]/#[[1]]}&/@Table[ {n,DivisorSigma[1,n]}, {n,10^6}],#[[3]]>1&],GreaterEqual[#1[[3]],#2[[3]]]&][[All,2]]] (* The program generates the first 31 terms of the sequence. *) (* Harvey P. Dale, Oct 04 2022 *)

a(n) = A000203(A004394(n)). - Amiram Eldar, Sep 25 2021

A153076 Smallest number m such that sigma(m) = A002110(n) where A002110(n) is the product of the first n primes.

Original entry on oeis.org

5, 29, 104, 1538, 13842, 188424, 3249576, 101864576, 2388809736, 59720243400, 2571228006912, 85266458294400, 6038197537693842, 227036227417289064, 12129959730964036050, 606111287080086323712, 36186481977293685109248

Offset: 2

Donovan Johnson, Dec 19 2008

nonn

			a(9) = 101864576. Sigma(101864576) = A002110(9) = 223092870 = 2*3*5*7*11*13*17*19*23.

Ray Chandler, Max Alekseyev, Table of n, a(n) for n = 2..24
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000203, A002110, A153077, A153078.

a(n) = A051444(A002110(n)). - Ray Chandler

Extended by Ray Chandler, Dec 28 2008

a(22)-a(24) added to b-file by Max Alekseyev, Jan 31 2012

A173015 Numbers k such that sequence of type a_k(n): {a(1) = 1, for n >= 2: a(n) = the smallest number h such that sigma(h) = A000203(h) = a(n-1) + k, a(n) = 0 if no such number exists} is sequence A063524(n) for n >= 1.

Original entry on oeis.org

9, 10, 16, 21, 22, 25, 26, 33, 34, 45, 46, 49, 50, 51, 52, 58, 64, 65, 66, 69, 70, 75, 76, 81, 82, 85, 86, 87, 88, 94, 99, 100, 105, 106, 115, 116, 117, 118, 122, 129, 130, 134, 135, 136, 141, 142, 145, 146, 147, 148, 153, 154, 165, 166, 169, 172, 177, 178, 184, 187, 188, 189, 190, 196

Offset: 1

Jaroslav Krizek, Nov 06 2010

nonn

Sequence of composite numbers.
A063524(n) = characteristic function of 1 = 1,0,0,0,0,0,0,0,0,0,0,0, ...
Numbers k such that A051444(k) = A051444(k+1) = 0.
Complement of A173016.

			a(1) = k = 9 because a_9(n) = A063524(n) = 1,0,0,0,0,0,0,0,0,0,0,0, ...

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

Cf. A000035, A000203, A051444, A063524, A173012, A173013, A173014, A173016.

seq[max_] := Module[{t = Table[1, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 0; SequencePosition[t, {0, 0}][[;; , 1]]]; seq[200] (* Amiram Eldar, Mar 22 2024 *)

More terms from Amiram Eldar, Mar 22 2024

A173016 Numbers k such that the sequence B = B_k defined by {B(1) = 1; for i >= 2: B(i) = the smallest number h such that sigma(h) = A000203(h) = B(i-1) + k; or B(i) = 0 if no such number h exists} is not the sequence {A063524(j): j >= 1}.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 17, 18, 19, 20, 23, 24, 27, 28, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 67, 68, 71, 72, 73, 74, 77, 78, 79, 80, 83, 84, 89, 90, 91, 92, 93, 95, 96, 97, 98, 101

Offset: 1

Jaroslav Krizek, Nov 06 2010

nonn

A063524(n) = characteristic function of 1 = 1,0,0,0,0,0,0,0,0,0,0,0, ...
Numbers k such that A051444(k) and A051444(k+1) are not simultaneously equal to 0.
Complement of A173015.

			a(1) = k = 1 because a_1(n)= A000035(n) = 1,0,1,0,1,0,1,0,1,0,1,0, ...
a(2) = k = 2 because a_2(n)= A173012(n) = 1,2,3,0,0,0,0,0,0,0,0,0, ...
a(3) = k = 3 because a_3(n)= A173013(n) = 1,3,5,7,0,2,0,2,0,2,0,2, ...
a(3) = k = 4 because a_4(n)= A173014(n) = 1,0,3,4,7,0,3,4,7,0,3,4, ...

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

Cf. A000035, A000203, A063524, A173012, A173013, A173014, A173015.

seq[max_] := Module[{t = Table[1, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 0; Complement[Range[max - 1], SequencePosition[t, {0, 0}][[;; , 1]]]]; seq[120] (* Amiram Eldar, Mar 22 2024 *)

Definition revised by Editors of OEIS, Mar 24 2024

A063972 a(n) is the smallest m such that usigma(m) = n (or 0 if no such m).

Original entry on oeis.org

1, 0, 2, 3, 4, 5, 0, 7, 8, 9, 0, 6, 0, 13, 0, 0, 16, 10, 0, 12, 0, 0, 0, 14, 0, 25, 0, 27, 0, 18, 0, 21, 32, 0, 0, 22, 0, 37, 0, 28, 0, 26, 0, 43, 0, 0, 0, 33, 0, 36, 0, 0, 0, 34, 0, 39, 0, 0, 0, 38, 0, 61, 0, 0, 64, 0, 0, 48, 0, 52, 0, 30, 0, 73, 0, 0, 0, 50, 0, 57, 0, 81, 0, 54, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Sep 05 2001

nonn

usigma(m) is the sum of unitary divisors of m, A034448.

			a(12)=6 because the unitary divisors of 6 are 1,2,3,6; they sum to 12.

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

Cf. A034444, A034448, A051444, A057637, A063973.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); m = 100; v = Table[0, {m}];  Do[u = usigma[n]; If[u <= m && v[[u]] == 0, v[[u]] = n], {n, 1, m}]; v (* Amiram Eldar, Sep 22 2019 *)

Edited by Don Reble, May 14 2006

cp's OEIS Frontend

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

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

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

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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.

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A007626 Sum of divisors of superabundant numbers (A004394).

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A153076 Smallest number m such that sigma(m) = A002110(n) where A002110(n) is the product of the first n primes.

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A173015 Numbers k such that sequence of type a_k(n): {a(1) = 1, for n >= 2: a(n) = the smallest number h such that sigma(h) = A000203(h) = a(n-1) + k, a(n) = 0 if no such number exists} is sequence A063524(n) for n >= 1.

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