A002191 -id:A002191 - cp's OEIS frontend

A352810 Values taken both by sigma (A000203) and by antisigma (A024816), where sigma is the sum of divisors function and antisigma is the sum of the non-divisors of n less than n function.

3, 20, 32, 54, 96, 132, 168, 217, 240, 252, 294, 338, 350, 464, 465, 582, 819, 1052, 1080, 1182, 1280, 1476, 1710, 1953, 2220, 2484, 2786, 3080, 3200, 3402, 3708, 4074, 4404, 4440, 4680, 4794, 5250, 5670, 6064, 6080, 6576, 6900, 7248, 7458, 8000, 8442, 8514, 8940

Offset: 1

Bernard Schott, Apr 04 2022

nonn

Common values attained by sigma and antisigma functions, in ascending order.
The asymptotic density of this sequence is 0 (according to 2nd comment of A002191).
The smallest integers k and m such that sigma(k) = antisigma(m) = a(n) are in A352811.

			As sigma(31) = 1+31 = 32 and antisigma(9) = 1+2+4+5+6+7+8 = 32, then 32 is a term.

Intersection of A002191 and A231365.

Cf. A000203, A024816, A352811.

m = 10^4; r = Range[m]; s = DivisorSigma[1, r]; as = r*(r + 1)/2 - s; Select[Intersection[s, as], # <= m &] (* Amiram Eldar, Apr 05 2022 *)

More terms from Amiram Eldar, Apr 05 2022

A352811 Table read by rows: row n gives triples (u, k, m) such that k and m are the smallest integers that respectively satisfy A352810(n) = u = A000203(k) = A024816(m).

Original entry on oeis.org

3, 2, 4, 20, 19, 7, 32, 21, 9, 54, 34, 11, 96, 42, 15, 132, 86, 18, 168, 60, 20, 217, 100, 22, 240, 114, 24, 252, 96, 23, 294, 164, 25, 338, 337, 27, 350, 349, 28, 464, 463, 31, 465, 200, 32, 582, 386, 35, 819, 288, 41, 1052, 1051, 48, 1080, 408, 47, 1182, 1181, 50

Offset: 1

Bernard Schott, Apr 12 2022

A000203 is the function sigma sum of divisors, while A024816 is the antisigma function, sum of the numbers less than n that do not divide n.

			The table begins:
  ------------------------------------------------------------------
  | row |      u =        | smallest k with  |    smallest m with  |
  |  n  |   A352810(n)    |  A000203(k) = u  |     A024816(m) = u  |
  ------------------------------------------------------------------
    n=1 :         3,                   2,                   4;
    n=2 :        20,                  19,                   7;
    n=3 :        32,                  21,                   9;
    n=4 :        54,                  34,                  11;
    n=5 :        96,                  42,                  15;
    n=6 :       132,                  86,                  18;
  ...................................................................
3rd row is (32, 21, 9) because A352810(3) = 32, sigma(21) = sigma(31) = 32 and antisigma(9) = 2+4+5+6+7+8 = 32, hence 21 and 9 are respectively the smallest integers k and m such that sigma(k) = antisigma(m) = 32.
5th row is (96, 42, 15) because A352810(5) = 96 and 42 and 15 are respectively the smallest integers k and m such that sigma(k) = antisigma(m) = 96.

Cf. A000203, A002191, A024816, A076617, A231365, A352810.

m = 2000; r = Range[m]; s = DivisorSigma[1, r]; as = r*(r + 1)/2 - s; i = Select[Intersection[s, as], # <= m &]; Flatten @ Transpose @ Join[{i}, Map[Flatten[Table[FirstPosition[#, i[[k]]], {k, 1, Length[i]}]] &, {s, as}]] (* Amiram Eldar, Apr 12 2022 *)

More terms from Amiram Eldar, Apr 13 2022

A354072 Perfect numbers that are the sum of the divisors of some number.

Original entry on oeis.org

6, 28, 496, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, 13164036458569648337239753460458722910223472318386943117783728128

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

The distinct values of A000203(A146542(n)).

Conjecture: 8128 is the only perfect number that is not in this sequence.

			The perfect number 28 is in the sequence because 28 = sigma(12).
sigma(727145809044307968) = sigma(1152771972099211264) = 2305843008139952128.

Amiram Eldar, Table of n, a(n) for n = 1..11

Intersection of A000396 and A002191.

Cf. A000203, A146542, A354073, A354074.

Set(Sort([&+Divisors(m): m in [1..10^7] | &+Divisors(&+Divisors(m)) eq 2 * &+Divisors(m)]))

a(8)-a(10) from Amiram Eldar, May 12 2024

A354073 Multiply-perfect numbers that are the sum of the divisors of some number.

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

Conjecture: 8128 is only multiply-perfect number that is not in this sequence.

The distinct values of A000203(A066961(n)).

			The multiply-perfect number 28 is in the sequence because 28 = sigma(12).

Amiram Eldar, Table of n, a(n) for n = 1..50

Intersection of A007691 and A002191.

Cf. A000203, A066961, A354072, A354074.

Set(Sort([&+Divisors(m): m in [1..10^7] | IsIntegral(&+Divisors(&+Divisors(m)) / &+Divisors(m))]))

a(18)-a(27) from Amiram Eldar, May 12 2024

A354074 Factorials that are the sum of the divisors of some m.

Original entry on oeis.org

1, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

Sequence of different values of A000203(A245015(n)).

Conjecture: number 2 is the only factorial that is not in this sequence.

			Number 24 is in the sequence because sigma(14) = sigma(15) = sigma(23) = 24.

Intersection of A000142 and A002191.

Cf. A000203, A338112, A055486, A055488, A245015, A354072, A354073, A338112.

More terms from Jinyuan Wang, May 17 2022

A090128 Distinct values of sigma(k), the sum of divisors, in order of appearance as k grows.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 28, 14, 24, 31, 39, 20, 42, 32, 36, 60, 40, 56, 30, 72, 63, 48, 54, 91, 38, 90, 96, 44, 84, 78, 124, 57, 93, 98, 120, 80, 168, 62, 104, 127, 144, 68, 126, 195, 74, 114, 140, 186, 121, 224, 108, 132, 180, 234, 112, 128, 252, 171, 156, 217

Offset: 1

Labos Elemer, Jan 16 2004

Constructed by reading A000203 and deleting values that already appeared earlier: A000203(15)=24 is dropped because equal to A000203(14). A000203(17)=18 is dropped because equal to A000203(10) etc. - R. J. Mathar, May 27 2024

Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A002191 (terms in ascending order).

Cf. A053212, A090127.

t = Table[DivisorSigma[1, w], {w, 100}]; u = Union[t]; uu = Union@ Table[ Min[ Flatten[ Position[t, u[[j]]]]], {j, Length[u]}]; Table[ t[[uu[[j]]]], {j, Length[uu]}]
DeleteDuplicates[DivisorSigma[1,Range[100]]] (* Harvey P. Dale, Dec 01 2018 *)

A124143 Perfect powers pp such that sigma(k) = pp for some positive integer k.

Original entry on oeis.org

4, 8, 32, 36, 121, 128, 144, 216, 256, 324, 400, 512, 576, 784, 900, 961, 1024, 1296, 1600, 1728, 1764, 1936, 2304, 2704, 2744, 2916, 3136, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5776, 5832, 6084, 6400, 7056, 7744, 7776, 8000, 8100, 8192, 9216, 9604

Offset: 1

Walter Kehowski, Dec 01 2006

nonn

			a(1) = 4 since sigma(3) = 4 = 2^2.

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

Intersection of A001597 and A002191 \ {1}.

Cf. A065496.

Set(Sort([SumOfDivisors(k): k in[1..10000], b in [2..15], a in [2..100] | SumOfDivisors(k) eq a^b])); // Jaroslav Krizek, Mar 10 2015

Set(Sort([SumOfDivisors(k): k in[A065496(n)]])); // Jaroslav Krizek, Mar 10 2015

with(numtheory); egcd := proc(n) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: P:={}: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; P:=P union {s}; fi od od; L; P;

powerQ[n_] := Block[{pf = FactorInteger@ n, min}, min = Min @@ Last /@ pf; min > 1 && AllTrue[Last /@ pf/min, IntegerQ]]; lim = 10000; Intersection[Select[Range@ lim, powerQ], DeleteDuplicates@ Sort[DivisorSigma[1, #] & /@ Range@ lim]] (* Michael De Vlieger, Mar 10 2015 *)

is(n) = ispower(n) && invsigmaNum(n) > 0; \\ Amiram Eldar, Aug 02 2024, using Max Alekseyev's invphi.gp

A161785 Numbers k that are in the range of both Euler's phi function and the sigma function.

Original entry on oeis.org

1, 4, 6, 8, 12, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 54, 56, 60, 72, 78, 80, 84, 96, 102, 104, 108, 110, 112, 120, 126, 128, 132, 138, 140, 144, 150, 156, 160, 162, 164, 168, 176, 180, 192, 198, 200, 204, 210, 212, 216, 222, 224, 228, 240, 252, 256, 260

Offset: 1

T. D. Noe, Jun 19 2009

nonn

That is, for each k there exist x and y such that k = phi(x) = sigma(y). Sigma is the sum of divisors function. Ford, Luca, and Pomerance prove that this sequence is infinite.

R. K. Guy, Unsolved Problems in Number Theory, B38.

T. D. Noe, Table of n, a(n) for n = 1..10000
Kevin Ford, Florian Luca, and Carl Pomerance, Common values of the arithmetic functions phi and sigma, Bull. London Math. Soc. 42 (2010), pp. 478-488.

Intersection[EulerPhi[Range[9660]], DivisorSigma[1,Range[2112]]]

list(lim)={
   my(u=vector(lim\=1,k,sigma(k)),v=vector(if(lim>63,3*lim*log(log(lim))\1,210),k,eulerphi(k)));
    select(n->n<=lim,setintersect(vecsort(v,,8),vecsort(u,,8)))
}; \\ Charles R Greathouse IV, Feb 05 2013

Intersection of A002202 and A002191.

A162967 Values taken by the sigma(sigma(n)) function A051027, with repetition, sorted into ascending order.

Original entry on oeis.org

1, 4, 7, 8, 12, 14, 15, 24, 24, 28, 28, 32, 32, 39, 39, 42, 56, 56, 60, 60, 60, 60, 63, 63, 72, 80, 84, 90, 91, 96, 96, 96, 96, 104, 112, 114, 120, 120, 120, 120, 124, 124, 124, 126, 128, 128, 133, 160, 168, 168, 168, 168, 171, 171, 186, 186, 195, 195, 195, 195, 195

Offset: 1

Jaroslav Krizek, Jul 19 2009

nonn

Removal of duplicates generates A070286. - R. J. Mathar, Jul 21 2009

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

Cf. A007609, A002191. - R. J. Mathar, Jul 21 2009

Cf. A051027, A070286.

f(k) = {my(v = invsigma(k), c = 0); for(i = 1, #v, c += invsigmaNum(v[i])); c;} \\ using Max Alekseyev's invphi.gp
list(lim) = {my(m); for(k = 1, lim, m = f(k); for(i = 1, m, print1(k, ", ")));} \\ Amiram Eldar, Dec 26 2024

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]]

cp's OEIS Frontend

A352810 Values taken both by sigma (A000203) and by antisigma (A024816), where sigma is the sum of divisors function and antisigma is the sum of the non-divisors of n less than n function.

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A352811 Table read by rows: row n gives triples (u, k, m) such that k and m are the smallest integers that respectively satisfy A352810(n) = u = A000203(k) = A024816(m).

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A354072 Perfect numbers that are the sum of the divisors of some number.

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Magma

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A354073 Multiply-perfect numbers that are the sum of the divisors of some number.

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Magma

Extensions

A354074 Factorials that are the sum of the divisors of some m.

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Crossrefs

Extensions

A090128 Distinct values of sigma(k), the sum of divisors, in order of appearance as k grows.

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Mathematica

A124143 Perfect powers pp such that sigma(k) = pp for some positive integer k.

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Magma

Magma

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PARI

A161785 Numbers k that are in the range of both Euler's phi function and the sigma function.

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