A258913 -id:A258913 - cp's OEIS frontend

A258914 Numbers k such that A258913(k) > k.

12, 18, 24, 31, 32, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 98, 104, 108, 114, 120, 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, 384

Offset: 1

Jeppe Stig Nielsen, Jun 14 2015

nonn

The first k of the form k = 6m-1 in this sequence is 86831.

What can be said about the asymptotic distribution of this set? If f(x) is the count of members not exceeding x, how does f behave as x tends to infinity?

Giovanni Resta, 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. A258913, A000203, A085790.

PARI
```
is_a(n)=sum(k=1,n,if(sigma(k)==n,k))>n
```

is(k) = vecsum(invsigma(k)) > k; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

A258968 a(n) is the least positive integer x with A054973(x) = n and A258913(x) < x.

Original entry on oeis.org

2, 3, 124, 10714158

Offset: 0

Jeppe Stig Nielsen, Jun 15 2015

Is a(n) well-defined for all n?
For n > 1, a(n) belongs to sequence A258931.
The terms a(k), for 4 <= k <= 100, if they exist, are larger than 3*10^10. - Giovanni Resta, Jun 15 2015

			For n=2, sigma(x)=124 for x=48 and 75, and 48+75 = 123 < 124.
For n=3, sigma(x)=10714158 for x=3031200, 3417300, and 3987450; and their sum is 10435950 (<10714158).

Cf. A054973, A258913, A258931, A007368, A258914.

a(n)=x=0;while(x++,u=List();for(i=1,x,if(sigma(i)==x,listput(u,i)));if(#u==n&vecsum(Vec(u))

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.

A258931 Numbers k such that card({x|sigma(x)=k}) > 1 and (Sum_{sigma(x)=k} x) < k.

Original entry on oeis.org

124, 378, 403, 1904, 3751, 4064, 5187, 5456, 6188, 9296, 9800, 11532, 12369, 13664, 14378, 15210, 16256, 16352, 17654, 18018, 18536, 19110, 19304, 19376, 20336, 21450, 22971, 23240, 23478, 24056, 24584, 24986, 25298, 26754, 28616, 28938, 31640, 33883, 34398

Offset: 1

Michel Marcus, Jun 15 2015

nonn

By definition these terms do not belong to A007370 nor to A007369.
All terms so far appear to be in A007371, with 2 pre-images. Are there any terms with more?
Yes, I find six up to 10^8 with 3 pre-images: 10714158, 12093224, 17315298, 30507906, 54891018, 81629262. - Charles R Greathouse IV, Jun 15 2015

			For k=124, the x's such that sigma(x)=124 are 48 and 75, and 48 + 75 = 123 < 124.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A159886.
Cf. A000203 (sigma, the sum of divisors), A085790.
Cf. A007369 (sigma(x)=n has no solution), A007370 (exactly 1 solution),
Cf. A007371 (exactly 2 solutions), A007372 (exactly has 3 solutions).
Cf. A258913 (Sum_{sigma(x)=n} x).

isok(n) = my(v = select(x->sigma(x)==n, vector(n, i, i))); (#v > 1) && (vecsum(v) < n);

list(lim)=my(v=vector(lim\1), u=List(), s); for(k=1,#v,s=sigma(k); if(s>#v,next); v[s]=if(v[s]==0, -k, abs(v[s])+k)); for(i=1,#v, if(v[i]>0 && v[i]Charles R Greathouse IV, Jun 15 2015

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A258914 Numbers k such that A258913(k) > k.

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A258968 a(n) is the least positive integer x with A054973(x) = n and A258913(x) < x.

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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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A258931 Numbers k such that card({x|sigma(x)=k}) > 1 and (Sum_{sigma(x)=k} x) < k.

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