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 A007369 Numbers n such that sigma(x) = n has no solution. (Formerly M1355) 42
 2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 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 empty sequence S. 2. 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 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 By analogy with the untouchable numbers (A005114), these numbers could be named "sigma-untouchable". - Daniel Lignon, Mar 28 2014 The asymptotic density of this sequence is 1 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020 REFERENCES 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. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS M. F. Hasler, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe.) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115. R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992 FORMULA A175192(a(n)) = 0, A054973(a(n)) = 0. - Jaroslav Krizek, Mar 01 2010 a(n) < 2n + sqrt(8n). - Charles R Greathouse IV, Oct 23 2015 EXAMPLE a(4) = 10 because there is no x < 10 whose sigma(x) = 10. MATHEMATICA a = {}; Do[s = DivisorSigma[1, n]; a = Append[a, s], {n, 1, 115} ]; Complement[ Table[ n, {n, 1, 115} ], Union[a] ] PROG (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 (PARI) A007369_list(LIM, m=0, L=List(), s)={for(n=2, LIM, (s=sigma(n-1))>LIM || bittest(m, s) || m+=1<

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Last modified September 24 20:13 EDT 2020. Contains 337321 sequences. (Running on oeis4.)