A007370 -id:A007370 - cp's OEIS frontend

A060665 Numbers k such that sigma(x) = k has exactly 9 solutions.

360, 480, 1488, 1800, 1824, 2184, 2232, 2640, 3120, 3420, 3696, 3744, 3960, 4200, 5292, 5580, 5808, 6144, 7344, 7980, 8100, 8352, 8448, 8784, 9144, 10164, 10296, 11592, 11664, 11970, 12432, 13968, 14520, 14560, 15504, 15600, 15912, 16224

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

Do we have a(n) ~ c*n where c ~= 700? - David A. Corneth, Sep 23 2019

			360 = sigma(120) = sigma(174) = sigma(184) = sigma(190) = sigma(267) = sigma(295) = sigma(319) = sigma(323) = sigma(359).

David A. Corneth, Table of n, a(n) for n = 1..10046 (first 8577 terms from Robert Israel, terms <= 7*10^6)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), this sequence (9), A060666 (10), A060678 (11), A060676 (12).

N:= 60000: # to get terms <= N
V:= Vector(N):
for k from 1 to N-1 do
  t:= numtheory:-sigma(k);
  if t <= N then V[t]:= V[t]+1 fi
od:
select(t -> V[t]=9, [$1..N]); # Robert Israel, Sep 22 2019

a = Table[ 0, {20000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 20001, a[ [ s ] ]++ ], {n, 1, 20000} ]; Select[ Range[ 20000 ], a[ [ # ] ] == 9 & ]

upto(n) = {my(v = vecsort(vector(n, i, sigma(i))), res = List()); for(i = 2, #v - 9, if(v[i-1] <= n && v[i-1] != v[i] && v[i] == v[i + 8] && v[i] != v[i+9], listput(res, v[i]))); res} \\ David A. Corneth, Sep 23 2019

is(k) = invsigmaNum(k) == 9 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060666 Numbers k such that sigma(x) = k has exactly 10 solutions.

Original entry on oeis.org

504, 864, 960, 1152, 1260, 2400, 3276, 3888, 4992, 6696, 7020, 7644, 8892, 9672, 9984, 11172, 11200, 11376, 11616, 11856, 12936, 13728, 13888, 14136, 14280, 15480, 15876, 15984, 17808, 19488, 21336, 22608, 23688, 24738, 24840, 25080

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			504 = sigma(204) = sigma(220) = sigma(224) = sigma(246) = sigma(284) = sigma(286) = sigma(334) = sigma(415) = sigma(451) = sigma(504).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), this sequence (10), A060678 (11), A060676 (12).

a = Table[ 0, {30000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 30001, a[ [ s ] ]++ ], {n, 1, 30000} ]; Select[ Range[ 30000 ], a[ [ # ] ] == 10 & ]

is(k) = invsigmaNum(k) == 10 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060676 Numbers k such that sigma (x) = k has exactly 12 solutions.

Original entry on oeis.org

1512, 1872, 2352, 3192, 3780, 4104, 4560, 4752, 5880, 6120, 8160, 8424, 8820, 11424, 13056, 15264, 16464, 16704, 17160, 17360, 17760, 18648, 19680, 19800, 20880, 22752, 23616, 24552, 24864, 27432, 30336, 30492, 31200, 32448, 35328

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			1512 = sigma(480) = sigma(636) = sigma(736) = sigma(748) = sigma(830) = sigma(902) = sigma(1006) = sigma(1105) = sigma(1255) = sigma(1391) = sigma(1411) = sigma(1511).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), this sequence (12).

a = Table[ 0, {50000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 50001, a[ [ s ] ]++ ], {n, 1, 50000} ]; Select[ Range[ 50000 ], a[ [ # ] ] == 12 & ]
Take[Sort[Transpose[Select[Tally[DivisorSigma[1,Range[100000]]],#[[2]] == 12&]][[1]]],50] (* Harvey P. Dale, Jan 18 2013 *)

is(k) = invsigmaNum(k) == 12 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060678 Numbers k such that sigma (x) = k has exactly 11 solutions.

Original entry on oeis.org

576, 1296, 2976, 3168, 3648, 3720, 4788, 4896, 5544, 6300, 9000, 9840, 10656, 11808, 12528, 13020, 13320, 14760, 15456, 16740, 17920, 18288, 18576, 19344, 19840, 20400, 21280, 22800, 23296, 24300, 26712, 26928, 27552, 27936, 28392

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			576 = sigma(210) = sigma(282) = sigma(310) = sigma(322) = sigma(345) = sigma(357) = sigma(382) = sigma(385) = sigma(497) = sigma(517) = sigma(527).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), this sequence (11), A060676 (12).

a = Table[ 0, {30000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 30001, a[ [ s ] ]++ ], {n, 1, 30000} ]; Select[ Range[ 30000 ], a[ [ # ] ] == 11 & ]

is(k) = invsigmaNum(k) == 11 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 1, 9, 13, 8, 0, 0, 1, 0, 19, 0, 0, 0, 1, 0, 0, 0, 12, 0, 29, 1, 1, 0, 0, 0, 22, 0, 37, 18, 27, 0, 1, 0, 43, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 49, 0, 0, 1, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 1, 0, 73, 0, 0, 0, 45, 0, 1, 0, 0

Offset: 1

Michel Marcus, Apr 10 2014

nonn

From n = 1 to 5, the least integers such that a(x) = n, depending on if singletons (see A007370 and A211656) are accepted or not, are 1, 3, 4, 7, 6 or 12, 126, 124, 210, 22152.

Is it possible to find an integer n such that a(n) = 6? Answer: n = A241625(6) = 6187272.

			There are no integers such that sigma(x) = 2, so a(2) = 0.
There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.
There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A007369, A007370, A211656, A241479 (a variant), A241625.

A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):
seq(A240667(n), n=1..82); # Peter Luschny, Apr 13 2014

a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];
Array[a, 100] (* Jean-François Alcover, Jul 30 2018 *)

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}

a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

a(A007369(n)) = 0.

A211657 Sigma(k) of numbers k such that value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 15, 13, 28, 14, 39, 20, 36, 40, 30, 63, 91, 38, 44, 78, 57, 93, 62, 127, 68, 195, 74, 121, 112, 171, 217, 102, 162, 110, 133, 255, 176, 160, 204, 138, 222, 266, 150, 300, 158, 363, 164, 183, 260, 174, 508, 194, 198, 200, 465, 306, 212, 256, 330

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

			For n = 4, a(n) = 7 because A211656(4) = 4; sigma (4) = 7.

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

Cf. A000203, A066076, A211656, A211658, A211659, A211660, A206036.

Cf. A007370 (sorted version of this sequence).

a(n) = sigma(A211656(n)).

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

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

A185147 Number of times each value of the sigma function occurs.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 3, 2, 2, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 3, 1, 1, 4, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 2, 1, 2, 2, 1, 1, 2, 5, 1, 2, 2, 1, 1, 1, 1, 6, 1, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 3, 3, 1, 3, 7, 1, 3, 6, 1, 1, 1, 1, 2, 1, 3, 2

Offset: 1

T. D. Noe, Mar 18 2011

nonn

The possible values of the sigma (sum of divisors) function are in A002191. Value A002191(n) occurs exactly a(n) times. Because sigma(x) >= x+1 (for x>1) with equality only at prime x, we know that for prime p, sigma(p) is the last time p+1 occurs as a value of sigma. This sequence is the same as A054973 without the zero terms.

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

Cf. A007370 (numbers for which a(n)=1).

Transpose[Sort[Tally[DivisorSigma[1, Range[Prime[PrimePi[200]]]]]]][[2]]

A332739 Numbers k such that usigma(x) = k has a unique solution, where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 14, 17, 26, 28, 33, 38, 40, 44, 56, 62, 65, 70, 74, 78, 82, 98, 100, 110, 112, 122, 129, 130, 136, 138, 158, 164, 174, 176, 182, 186, 190, 194, 208, 210, 212, 220, 222, 230, 238, 242, 244, 246, 248, 250, 256, 257, 258, 278, 282, 284, 290

Offset: 1

Amiram Eldar, Feb 21 2020

nonn

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

Cf. A007370, A034448, A064000.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); m = 300; v = Table[ 0, {m}]; Do[u = usigma[k]; If[u <= m, v[[u]]++], {k, 1, m}]; Position[v, _?(# == 1 &)]//Flatten

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A060665 Numbers k such that sigma(x) = k has exactly 9 solutions.

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A060666 Numbers k such that sigma(x) = k has exactly 10 solutions.

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A060676 Numbers k such that sigma (x) = k has exactly 12 solutions.

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A060678 Numbers k such that sigma (x) = k has exactly 11 solutions.

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A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

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A211657 Sigma(k) of numbers k such that value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

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

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