A007369 - OEIS (original) (raw)

A007369

Numbers n such that sigma(x) = n has no solution.
(Formerly M1355)

51

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

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 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

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. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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.

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<<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

CROSSREFS

See A083532 for the gaps, i.e., first differences.

See A048995 for the missed sums of nontrivial divisors.