A085790 -id:A085790 - cp's OEIS frontend

A007609 Values taken by the sigma function A000203, listed with multiplicity and in ascending order.

1, 3, 4, 6, 7, 8, 12, 12, 13, 14, 15, 18, 18, 20, 24, 24, 24, 28, 30, 31, 31, 32, 32, 36, 38, 39, 40, 42, 42, 42, 44, 48, 48, 48, 54, 54, 56, 56, 57, 60, 60, 60, 62, 63, 68, 72, 72, 72, 72, 72, 74, 78, 80, 80, 84, 84, 84, 90, 90, 90, 91, 93, 96, 96, 96, 96, 98, 98

Offset: 1

Walter Nissen

A175192(a(n)) = 1, A054973(a(n)) >= 1. - Jaroslav Krizek, Mar 01 2010

a(n) is the median of the values of A000203(m) from m=1 to m=2n-1. (This needs confirmation as it relies on the assumption that A000203(n) is always bigger than the median of the values A000203(x) from x=1 to x=n.) - Chayim Lowen, May 27 2015

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A002191 (duplicates removed), A007368, A085790.

sort(select(`<=`,map(numtheory:-sigma,[$1..1000]),1001)); # Robert Israel, Jun 04 2015

terms = 68; ClearAll[t]; t[k_] := t[k] = Sort[ Table[ DivisorSigma[1, n], {n, 1, k*terms}]][[1 ;; terms]]; t[k = 2]; While[t[k] != t[k-1], k++]; t[k] (* Jean-François Alcover, Nov 21 2012 *)
With[{nn=80},Take[Sort[DivisorSigma[1,Range[nn*100]]],nn]] (* Harvey P. Dale, Mar 09 2016 *)

list(lim)=select(k->k<=lim,Set(apply(sigma,[1..lim\1]))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A085790(n)). - Jinyuan Wang, Apr 15 2020

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 *)

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.

A263025 n is the a(n)-th positive integer having its sum of divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 5, 1, 1, 1, 2, 1, 4, 2, 2, 1, 1, 2, 3, 1, 1, 1, 3

Offset: 1

Paul Tek, Oct 09 2015

Sum of divisors is given by A000203.
This can also be described as the ordinal transform of A000203. - Franklin T. Adams-Watters, Oct 09 2015
a(n) > 1 iff n is in A069822.

			The numbers with sum of divisors 72 are: 30, 46, 51, 55, 71.
Hence: a(30)=1, a(46)=2, a(51)=3, a(55)=4, a(71)=5.
More generally: the terms of each row of A085790 (say of length i) map to 1, 2, ..., i.
Also: for any n>0, the n terms of the n-th row of A201915 map to 1, 2, ..., n.

Paul Tek, Table of n, a(n) for n = 1..25000

Cf. A000203, A034885, A069822, A078898, A085790, A201915.

N:= 1000: # to get a(1) to a(N)
Sigmas:= [seq(numtheory:-sigma(i),i=1..N)]:
seq(numboccur(Sigmas[n], Sigmas[1..n]),n=1..N); # Robert Israel, Oct 09 2015

t = DivisorSigma[1, #] & /@ Range@ 10000; s = Position[t, #] & /@ Range@ Max@ t; Flatten[Position[s, #, {3}]][[2]] & /@ Range@ 87 (* Michael De Vlieger, Oct 09 2015 *)

cnt = vector(224); for (n=1, 87, s=sigma(n); cnt[s] = cnt[s]+1; print1(cnt[s] ", "))

a(A034885(k))=1 for k>0.

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

Original entry on oeis.org

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

A206030 Numbers m with at least two divisors d with the same sigma(d).

Original entry on oeis.org

66, 132, 170, 198, 210, 260, 264, 322, 330, 340, 345, 396, 400, 420, 456, 462, 510, 520, 528, 594, 630, 644, 651, 660, 680, 690, 726, 780, 792, 800, 820, 840, 850, 858, 912, 924, 966, 990, 1020, 1035, 1040, 1050, 1056, 1066, 1092, 1122, 1155, 1160, 1188

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Complement of sequence contains numbers whose divisors d have distinct values of sigma(d).

			66 is in sequence because two divisors d (6 and 11) of 66 have the same sigma(d) = 12.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000203, A007609, A085790.

Select[Range[1200], Length[DivisorSigma[1, Divisors[#]]] != Length[Union[DivisorSigma[1, Divisors[#]]]] &] (* T. D. Noe, Feb 10 2012 *)
[Range[1200],Max[Tally[DivisorSigma[1,Divisors[#]]][[;;,2]]]>1&] (* Harvey P. Dale, Sep 27 2024 *)

ok(n)={my(v=apply(sigma, divisors(n))); #Set(v) < #v} \\ Andrew Howroyd, Aug 01 2018

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

Original entry on oeis.org

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

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

A342555 2*a(n) is the start of 3 consecutive numbers (even-odd-even) that are sums of divisors, i.e., terms of A000203.

Original entry on oeis.org

3, 6, 15, 19, 63, 153, 199, 255, 423, 480, 511, 546, 861, 1111, 1189, 1400, 1770, 1875, 1935, 1995, 2047, 2556, 3475, 3619, 4005, 4095, 4920, 5151, 5215, 6649, 8046, 8191, 8646, 8749, 9765, 11175, 11199, 14028, 14197, 15391, 15427, 15470, 16383, 19494, 25878, 26557, 26799

Offset: 1

Hugo Pfoertner, May 14 2021

nonn

There exists one exceptional case of 4 consecutive numbers 12, 13, 14, 15, where 13 would start an odd-even-odd progression.

			a(1) = 3, because 2*3 = 6 is the start of the first occurrence of a row of 3 consecutive numbers, all of which are in A000203. 6 = sigma(5), 7 = sigma(4), 8 = sigma(7).
a(2) = 6: 2*6 = 12 = sigma(6) = sigma(11), 13 = sigma(9), 14 = sigma(13). 15 = sigma(8), which would be at the end of the row 13, 14, 15, is excluded by the even-odd-even condition.
a(3) = 15: 2*15 = 30 = sigma(29), 31 = sigma(16) = sigma(25), 32 = sigma(21) = sigma(31).
See Jeppe Stig Nielsen's list for more examples.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005.
Jeppe Stig Nielsen, List of numbers with divisor sum k, k<=10000.

Cf. A000203, A085790, A342560.

a342555(nterms) = {my(N=vector(3, i, invsigmaNum(i+1)), n=0, k=4); while(n<=nterms, if(vecmin(N)>0 && !(k%2), print1((k-2)/2, ", "); n++); k++; N[1+k%3] = invsigmaNum(k))}; \\ see Alekseyev link for invsigmaNum()
a342555(46)

A342560 2*a(n) is the first of 5 consecutive even numbers that are sums of divisors, i.e., terms of A000203.

Original entry on oeis.org

18, 78, 270, 306, 558, 846, 1098, 1182, 1188, 1590, 1608, 1626, 2106, 2196, 2298, 2538, 2718, 2868, 4368, 4590, 4716, 4806, 4926, 4950, 6078, 7866, 8646, 8700, 8952, 9150, 9558, 9918, 10176, 10506, 10998, 11358, 11778, 12648, 12870, 13116, 13530, 13638, 15090, 16806

Offset: 1

Hugo Pfoertner, May 11 2021

nonn

			a(1) = 18, because 2*18 = 36 is the first occurrence of a row of 5 consecutive even numbers, all of which are in A000203. 36 = sigma(22), 38 = sigma(37), 40 = sigma(27), 42 = sigma(20) = sigma(26) = sigma(41), 44 = sigma(43);
a(2) = 78: 2*78 = 156 = sigma(99) = sigma(125), 158 = sigma(157), 160 = sigma(133), 162 = sigma(106), 164 = sigma(163);
See Jeppe Stig Nielsen's list for more examples.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005.
Jeppe Stig Nielsen, List of numbers with divisor sum k, k<=10000.

Cf. A000203, A085790, A342555.

a342560(nterms) = {my(N=vector(5,i,invsigmaNum(2*i)),n=0,k=10,j=4); while(n<=nterms, if(vecmin(N)>0,print1((k-8)/2,", "); n++); k+=2; N[1+(j++)%5] = invsigmaNum(k))};
a342560(49) \\ see Alekseyev link for invsigmaNum()

cp's OEIS Frontend

A007609 Values taken by the sigma function A000203, listed with multiplicity and in ascending order.

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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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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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A263025 n is the a(n)-th positive integer having its sum of divisors.

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

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A206030 Numbers m with at least two divisors d with the same sigma(d).

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

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