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A007369 Numbers n such that sigma(x) = n has no solution.

%I A007369 M1355 #72 Nov 08 2024 08:47:53
%S A007369 2,5,9,10,11,16,17,19,21,22,23,25,26,27,29,33,34,35,37,41,43,45,46,47,
%T A007369 49,50,51,52,53,55,58,59,61,64,65,66,67,69,70,71,73,75,76,77,79,81,82,
%U A007369 83,85,86,87,88,89,92,94,95,97,99,100,101,103,105,106,107,109,111,113
%N A007369 Numbers n such that sigma(x) = n has no solution.
%C A007369 With an initial 1, may be constructed inductively in stages from the list L = {1,2,3,....} by the following sieve procedure. Stage 1. Add 1 as the first term of the sequence a(n) and strike off 1 from L. Stage n+1. Add the first (i.e. leftmost) term k of L as a new term of the sequence a(n) and strike off k, sigma(k), sigma(sigma(k)),.... from L. - _Joseph L. Pe_, May 08 2002
%C A007369 This sieve is a special case of a more general sieve. Let D be a subset of N and let f be an injection on D satisfying f(n) > n. Define the sieve process as follows: 1. Start with the empty sequence S and let E = D. 2. Append the smallest element s of E to S. 3. Remove s, f(s), f(f(s)), f(f(f(s))), ... from E. 4. Go to step 2. After this sieving process, S = D - f(D). To get the current sequence, take f = sigma and D = {n | n >= 2}. - _Max Alekseyev_, Aug 08 2005
%C A007369 By analogy with the untouchable numbers (A005114), these numbers could be named "sigma-untouchable". - _Daniel Lignon_, Mar 28 2014
%C A007369 The asymptotic density of this sequence is 1 (Niven, 1951, Rao and Murty, 1979). - _Amiram Eldar_, Jul 23 2020
%D A007369 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A007369 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007369 M. F. Hasler, <a href="/A007369/b007369.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A007369 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A007369 Ivan Niven, <a href="https://doi.org/10.1090/S0002-9904-1951-09543-9">The asymptotic density of sequences</a>, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
%H A007369 R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, <a href="https://doi.org/10.4153/CMB-1979-018-5">On a theorem of Niven</a>, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
%H A007369 R. G. Wilson, V, <a href="/A007015/a007015.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>
%F A007369 A175192(a(n)) = 0, A054973(a(n)) = 0. - _Jaroslav Krizek_, Mar 01 2010
%F A007369 a(n) < 2n + sqrt(8n). - _Charles R Greathouse IV_, Oct 23 2015
%e A007369 a(4) = 10 because there is no x < 10 whose sigma(x) = 10.
%t A007369 a = {}; Do[s = DivisorSigma[1, n]; a = Append[a, s], {n, 1, 115} ]; Complement[ Table[ n, {n, 1, 115} ], Union[a] ]
%o A007369 (PARI) list(lim)=my(v=List(),u=vectorsmall(lim\1),t); for(n=1,lim, t=sigma(n); if(t<=lim, u[t]=1)); for(n=2,lim, if(u[n]==0, listput(v,n))); Vec(v) \\ _Charles R Greathouse IV_, Mar 09 2017
%o A007369 (PARI) A007369_list(LIM,m=0,L=List(),s)={for(n=2,LIM,(s=sigma(n-1))>LIM || bittest(m,s) || m+=1<<s; bittest(m,n)||listput(L,n));L} \\ A bit slower, but bitmask requires less memory, avoiding stack overflow produced by the earlier code for lim = 1e6 with standard gp setup. - _M. F. Hasler_, Mar 12 2018
%Y A007369 Complement of A002191.
%Y A007369 See A083532 for the gaps, i.e., first differences.
%Y A007369 See A048995 for the missed sums of nontrivial divisors.
%K A007369 nonn
%O A007369 1,1
%A A007369 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E A007369 More terms from _David W. Wilson_