A300869 -id:A300869 - cp's OEIS frontend

A159886 Values k such that sigma(x) = k has more than one solution, sigma = A000203.

12, 18, 24, 31, 32, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 98, 104, 108, 114, 120, 124, 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, 378, 384, 390

Offset: 1

Jaroslav Krizek, Apr 25 2009

nonn

Numbers k with A054973(k) >= 2. Numbers k which occur in A000203 more than once.
Numbers k = A007609(n) with A007609(n+1) - A007609(n) = 0.
Does this sequence have finite density? - Franklin T. Adams-Watters, Jun 18 2009
See A300869 for the odd terms, much less frequent since they can only occur for x = k^2 or 2*k^2. - M. F. Hasler, Mar 16 2018

			a(1) = 12 as the multiplicity of the value 12 is 2: 12 = sigma(6) = sigma(11).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1095 from Franklin T. Adams-Watters)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A054973, A007609, A007368.
Subsequence of A002191.
Odd terms are listed in A300869.

na(n) = local(v, s); v=vector(n);for(k=1,n,s=sigma(k);if(s<=n,v[s]++));v
la(n) = local(v, r); v=na(n);r=[];for(k=1,n,if(v[k]>1,r=concat(r,[k])));r \\ Franklin T. Adams-Watters, Jun 18 2009

is(k) = invsigmaNum(k) > 1; \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

Edited and extended by R. J. Mathar, Apr 28 2009

A123523 Smallest odd number k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

1, 31, 347529, 10773399, 4104665019, 77253471477, 28732655133, 35492068813383, 108695634368139, 461396894573979, 68452476460273269, 2529134502772059, 99414440839732473

Offset: 1

T. D. Noe, Oct 02 2006

Note that sigma(x) is odd iff x is in A028982 (numbers of the form m^2 or 2m^2 for m > 0).
a(14) > 10^18. a(15) = 175792216832685999. a(16) > 10^18. - Donovan Johnson, Jun 09 2011
From David A. Corneth, Apr 27 2019: (Start)
The least common divisor of the first 13 terms is k = 63540409508528099686942221. Checking the divisors of k to see if they give an upper bound for some a(n) gives these upper bounds:
a(14) <= 2489145199534927711323, for n = 16..27, a(n) <= 30520233337797869211, 1292387730916522149, 3939513268555279291149, 1066776514086397590567, 7538497634436073695117, 1629700928685734429889, 7217246969893966760937, 136456488459785229549035859, 396763033391372299743, 2215694819757447795607659, 500318185106520469975923, 5916133590898752361467873 respectively.
All these listed upper bounds are divisors of 12302819034343122006137404371659222028537. No more divisors of this number are an upper bound for any n.
This method doesn't give a stronger lower bound except that it tells us that a new upper bound for some term is no divisor of k. (End)

			For n = 3, sigma(x) = 347529 has exactly three solutions x = 164836, 203522, 239121.

Max Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions, Journal of Integer Sequences 19 (2016), Article 16.5.2
David A. Corneth, PARI program, partially written by Max Alekseyev

Cf. A007368, A300869.

Different from A123524.

Needs["Statistics`DataManipulation`"]; nn=10^6; t1=DivisorSigma[1,Range[nn]^2]; t2=DivisorSigma[1,2*Range[nn/Sqrt[2]]^2]; t=Join[t1,t2]; {u,v}=Transpose[Sort[Frequencies[t]]]; Table[p=Position[u,i,1,1][[1,1]]; v[[p]], {i,Length[Union[u]]}]

a(8) from Martin Fuller, Oct 07 2006
a(9)-a(10) from Donovan Johnson, Dec 09 2008
a(11)-a(13) from Donovan Johnson, Jun 09 2011

A331036 Odd values of the sum-of-divisors function sigma (A000203), listed by increasing size and with multiplicity.

Original entry on oeis.org

1, 3, 7, 13, 15, 31, 31, 39, 57, 63, 91, 93, 121, 127, 133, 171, 183, 195, 217, 255, 307, 363, 381, 399, 399, 403, 403, 465, 511, 549, 553, 741, 781, 819, 847, 855, 871, 921, 931, 961, 993, 1023, 1093, 1143, 1209, 1281, 1407, 1651, 1659, 1723, 1729, 1767, 1767, 1815, 1893, 1953

Offset: 1

M. F. Hasler, Jan 08 2020

nonn

See A060657 for the range (without repeated terms) and A152677 for the subsequence of odd values in A000203.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060657, A023195 (subset of primes), A152677 (subsequence of odd values in A000203), A300869 (repeated terms).

N:= 2000: # for terms <= N
Res:= NULL:
for m from 1 to floor(sqrt(N)) by 2 do
  sm:= numtheory:-sigma(m^2);
  for k from 1 to floor(log[2](N/sm+1)) do
    v:= sm*(2^k-1);
    if v <= N then Res:= Res, v; count:= count+1 fi;
  od
od:
sort([Res]); # Robert Israel, Jan 14 2020

Sort@ Select[DivisorSigma[1, Range@ 2000], OddQ[#] && # < 2000 &] (* Giovanni Resta, Jan 08 2020 *)

list(lim)=select(k->k<=lim, vecsort(apply(sigma, concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2))))) \\ Charles R Greathouse IV, Feb 15 2013 [originally added in A152677]

A300779 Odd numbers x such that x and x + 2 are both sums of divisors, i.e., elements of A000203.

Original entry on oeis.org

1, 13, 91, 241573, 38152387, 139415801707, 55342019130181, 61166380109329, 417542026135897, 417542026135897, 13805828672331787

Offset: 1

Hugo Pfoertner, Mar 12 2018

If some x or x + 2 is in A300869, i.e., it has more than one representation as sigma(m), as for x = 417542026135897 = sigma((4*17*209459)^2) = sigma((5*17*209459)^2) = sigma((2*7723267)^2) - 2, then it is listed with multiplicity and all corresponding pairs of numbers are provided in A300780.

			a(1) = 1 because 1 = sigma(1) and 3 = sigma(2),
a(2) = 13: 13 = sigma(9) and 15 = sigma(8),
a(3) = 91: 91 =sigma(36), 93 = sigma(50),
a(4) = 241573: 241573 = sigma(241081), 241575 = sigma(117128),
a(5) = 38152387: 38152387 = sigma(15069924), 38152389 = sigma(23011209).

Seqfan Mailing List Thread, Consecutive sigmas, with contributions from _Franklin T. Adams-Watters_, _Hugo Pfoertner_ and _Emmanuel Vantieghem_.

Cf. A000203, A002191, A083531, A300780 (numbers corresponding to sigma values), A300869.

a(5) from Emmanuel Vantieghem

a(6)-a(11) from Giovanni Resta, Mar 13 2018

cp's OEIS Frontend

A159886 Values k such that sigma(x) = k has more than one solution, sigma = A000203.

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A123523 Smallest odd number k such that sigma(x) = k has exactly n solutions.

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A331036 Odd values of the sum-of-divisors function sigma (A000203), listed by increasing size and with multiplicity.

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Maple

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A300779 Odd numbers x such that x and x + 2 are both sums of divisors, i.e., elements of A000203.

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