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%I M2318 N0916 #67 Oct 14 2023 16:05:00
%S 1,3,4,6,7,8,12,13,14,15,18,20,24,28,30,31,32,36,38,39,40,42,44,48,54,
%T 56,57,60,62,63,68,72,74,78,80,84,90,91,93,96,98,102,104,108,110,112,
%U 114,120,121,124,126,127,128,132,133,138,140,144,150,152,156
%N Possible values for sum of divisors of n.
%C Distinct values attained by the sigma(n) function, in ascending order.
%C The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - _Amiram Eldar_, Jul 23 2020
%D J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Giovanni Resta, <a href="/A002191/b002191.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H 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, p. 840.
%H 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 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.
%F a(n)/n < log_10(n) + O(1) with O(1) <= 1 for all n. - _M. F. Hasler_, Nov 22 2019
%e a(100) = 272, a(10^3) = 3696, a(10^4) = 44496, a(10^5) = 510356, a(10^6) = 5691216. - _M. F. Hasler_, Nov 22 2019
%p N:= 1000: # to get all entries <= N
%p select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N); # _Robert Israel_, Jun 16 2014
%t lim=1000; Select[Union[DivisorSigma[1,Range[lim]]], #<=lim &] (* _T. D. Noe_, May 06 2010 *)
%o (PARI) list(lim)=select(n->n<=lim,Set(vector(lim\=1,n,sigma(n)))) \\ _Charles R Greathouse IV_, Nov 12 2013
%o (PARI) A002191_upto(N,M=N\1+1)=Set(apply(t->min(sigma(t),M), [1..N\1-1]))[^-1] \\ Needs big stack for N >= 10^6; slower alternative: {A002191_upto(N)= my(L=List(1),s); for(n=2,N\=1,N<(s=sigma(n))||listput(L,s));Set(L)}
%o A2191=A002191_upto(1e4); A002191(n)={#A2191<n&& A2191=A002191_upto(n*logint(n,10)+n); A2191[n]} \\ - _M. F. Hasler_, Nov 22 2019
%Y Cf. A000203, A007609.
%Y Complement of A007369. A175192(a(n)) = 1, A054973(a(n)) >= 1. - _Jaroslav Krizek_, Mar 01 2010
%Y See A083531 for the gaps, i.e., first differences. - _M. F. Hasler_, Mar 12 2018
%Y Subsequence of A211347.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
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