A002191 Possible values for sum of divisors of n.

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, 56, 57, 60, 62, 63, 68, 72, 74, 78, 80, 84, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 124, 126, 127, 128, 132, 133, 138, 140, 144, 150, 152, 156

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A000203, A007609.

Complement of A007369. A175192(a(n)) = 1, A054973(a(n)) >= 1. - Jaroslav Krizek, Mar 01 2010

See A083531 for the gaps, i.e., first differences. - M. F. Hasler, Mar 12 2018

Subsequence of A211347.

Programs

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 *)

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)={#A2191A002191_upto(n*logint(n,10)+n); A2191[n]} \\ - M. F. Hasler, Nov 22 2019

Formula

a(n)/n < log_10(n) + O(1) with O(1) <= 1 for all n. - M. F. Hasler, Nov 22 2019