A002191 Possible values for sum of divisors of n. (Formerly M2318 N0916)

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

1,2

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

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.

Sequence in context: A066650 A007609 A285703 * A108348 A085149 A239458

Adjacent sequences: A002188 A002189 A002190 * A002192 A002193 A002194

Keyword

nonn

Author

N. J. A. Sloane

Status

approved