OFFSET
1,2
COMMENTS
Distinct values attained by the sigma(n) function, in ascending order.
The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020
REFERENCES
J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Giovanni Resta, 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, p. 840.
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.
FORMULA
a(n)/n < log_10(n) + O(1) with O(1) <= 1 for all n. - M. F. Hasler, Nov 22 2019
EXAMPLE
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
MAPLE
N:= 1000: # to get all entries <= N
select(`<=`, {seq(numtheory[sigma](i), i=1..N)}, N); # Robert Israel, Jun 16 2014
MATHEMATICA
lim=1000; Select[Union[DivisorSigma[1, Range[lim]]], #<=lim &] (* T. D. Noe, May 06 2010 *)
PROG
(PARI) list(lim)=select(n->n<=lim, Set(vector(lim\=1, n, sigma(n)))) \\ Charles R Greathouse IV, Nov 12 2013
(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)}
A2191=A002191_upto(1e4); A002191(n)={#A2191<n&& A2191=A002191_upto(n*logint(n, 10)+n); A2191[n]} \\ - M. F. Hasler, Nov 22 2019
CROSSREFS
See A083531 for the gaps, i.e., first differences. - M. F. Hasler, Mar 12 2018
Subsequence of A211347.
KEYWORD
nonn
AUTHOR
STATUS
approved