A054973 -id:A054973 - cp's OEIS frontend

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

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

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

A332036 Number of integers whose bi-unitary divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 1, 1, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(12) = 2 since there are 2 solutions to bsigma(x) = 12 (bsigma is A188999): 6 and 11.

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

Cf. A054973, A063974, A188999, A332038, A332040.

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]); m = 100; v = Table[0, {m}]; Do[b = bsigma[k]; If[b <= m, v[[b]]++], {k, 1, m}]; v

A332038 Number of integers whose infinitary divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(12) = 2 since there are 2 solutions to isigma(x) = 12 (isigma is A049417): 6 and 11.

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

Cf. A054973, A049417, A063974, A332036, A332040.

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]); m = 100; v = Table[0, {m}]; Do[i = isigma[k]; If[i <= m, v[[i]]++], {k, 1, m}]; v

A332040 Number of integers whose exponential divisors sum to n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 5, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 3, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(6) = 2 since there are 2 solutions to esigma(x) = 6 (esigma is A051377): 4 and 6.

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

Cf. A054973, A051377, A063974, A332036, A332038.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; m = 100; v = Table[0, {m}]; Do[e = esigma[k]; If[e <= m, v[[e]]++], {k, 1, m}]; v

A111865 Expansion of g.f. Product_{k>=1} 1/(1-x^sigma(k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 9, 11, 14, 17, 24, 29, 36, 46, 57, 66, 85, 103, 125, 151, 182, 213, 264, 310, 368, 440, 524, 604, 724, 849, 998, 1164, 1363, 1573, 1854, 2136, 2481, 2879, 3336, 3807, 4427, 5079, 5844, 6698, 7695, 8754, 10072, 11451, 13075, 14898, 16988

Offset: 0

Jon Perry, Nov 23 2005

nonn

Number of partitions of n into parts of size p = sigma(k) for some k, when there are A054973(p) kinds of part p.

			a(6) = 5 : We have sigma(1)=1, sigma(2)=3, sigma(3)=4, sigma(5)=6 so 111111, 1113, 114, 6 and 33.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Cf. A000203, A002191, A007609, A054973, A305320.

with(numtheory):
seq(coeff(series(mul(1/(1-x^sigma(k)),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, May 31 2018

CoefficientList[ Series[Product[1/(1 - x^DivisorSigma[1, k]), {k, 47}], {x, 0, 52}], x] (* Robert G. Wilson v, Nov 25 2005 *)

lista(nn) = Vec(prod(k=1, nn, 1/(1-x^sigma(k))+ O(x^nn))) \\ Michel Marcus, May 30 2018

G.f.: Product_{k>=1} 1/(1-x^sigma(k)).

More terms from Robert G. Wilson v, Nov 25 2005

a(0)=1 prepended by Seiichi Manyama, May 30 2018

A206027 a(n) = the number of solutions to sigma(x) = A145899(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 15, 21, 29, 35, 37, 49, 58, 59, 68, 79, 98, 111, 160, 224, 256, 317, 382, 426, 431, 563, 576, 624, 774, 865, 883, 1050, 1195, 1265, 1371, 1376, 1742, 1755, 1935, 2095, 2437, 2447, 2944, 3055, 3318, 3324, 3366, 4289, 4369, 4502

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

			a(4) = 5 because the 5 numbers x such that sigma(x) = A145899(4) = 72 are x = 30, 46, 51, 55, 71.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000203 (sigma = sum of divisors of n), A145899, A206026.

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]][[2]]]], {n, Length[t2]}]; t3 (* T. D. Noe, Feb 03 2012 *)

a(n) = A054973(A145899(n)). - Michel Marcus, Oct 22 2013

Extended to 1376 by T. D. Noe, Feb 04 2012

Terms a(41) and beyond from Donovan Johnson, Feb 04 2012

A153078 Number of values of m such that sigma(m) = A002110(n) where A002110(n) is the product of the first n primes.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 2, 4, 5, 3, 7, 5, 10, 2, 8, 4, 5, 6, 11, 32, 42, 68, 24, 87

Offset: 1

Donovan Johnson, Dec 19 2008

			a(10) = 3 because 2388809736, 3450503048 and 3696967556 are the only numbers with a sigma value = A002110(10). A002110(10) = 6469693230 = 2*3*5*7*11*13*17*19*23*29.

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, A153076, A153077.

a(n) = A054973(A002110(n)). - Ray Chandler, Dec 28 2008

a(12)-a(21) from Ray Chandler, Dec 28 2008

a(22)-a(24) from Max Alekseyev, Jan 27 2012

A175253 a(n) = characteristic function of numbers k such that A000203(m) = k has no solution for any m, where A000203(m) = sum of divisors of m.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Mar 14 2010

nonn

a(n) = characteristic function of numbers from A007369(n). a(n) = 1 if A000203(m) not equal to n for any m, else 0. a(n) = 1 for such n that A054973(n) = 0. a(n) = 0 for such n that A054973(n) >= 1. a(n) + A175192(n) = A000012(n).

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A074753, A202275, A202276, A175523, A007369, A007368, A007609.

seq[max_] := Module[{t = Table[0, {max}]}, t[[Complement[Range[max], Table[ DivisorSigma[1, n], {n, 1, max}]]]] = 1; t]; seq[100] (* Amiram Eldar, Mar 22 2024 *)

More terms from Jaroslav Krizek, Dec 25 2011.

A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.

Original entry on oeis.org

1, 12, 24, 72, 72, 168, 240, 336, 360, 504, 576, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 4320, 4320, 4320, 4320, 4320, 4320, 5760, 5760, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence of numbers from A145899.

			a(6) = 168 because 168 is the smallest value of sigma(k) for n = 6 positive integers k such that sigma(k) = 168 has solution; k = 60, 78, 92, 123, 143, 167.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A085790, A145899, A206027.

list(len) = {my(v = vector(len), k = 1, c = 0, i); while(c < len, i = invsigmaNum(k); for(j = 1, i, if(j <= len && v[j] == 0, v[j] = k; c++)); k++); v;} \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

cp's OEIS Frontend

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

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A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

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A332036 Number of integers whose bi-unitary divisors sum to n.

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Mathematica

A332038 Number of integers whose infinitary divisors sum to n.

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A332040 Number of integers whose exponential divisors sum to n.

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A111865 Expansion of g.f. Product_{k>=1} 1/(1-x^sigma(k)).

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A206027 a(n) = the number of solutions to sigma(x) = A145899(n).

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A153078 Number of values of m such that sigma(m) = A002110(n) where A002110(n) is the product of the first n primes.

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