OFFSET
1,1
COMMENTS
Note that a(32) = 125038913126400 = 11182080^2. - Zak Seidov, Aug 29 2012
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..1500 (first 200 terms from T. D. Noe)
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
Graeme L. Cohen, Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100.
R. G. Wilson, V, Notes, n.d.
FORMULA
Conjecture: (1/2)*log(n) < a(n+1)/a(n) < 2*log(n). - Benoit Cloitre, May 08 2003
Conjecture: a(n) == 0 mod 9 for n > 34. - Ivan N. Ianakiev, Mar 27 2014
Checked up to n = 1000. Similar statements hold for other small primes. For example, a(n) seems to be divisible by 2^22 * 3^5 * 5 * 7 = 35672555520 for all n > 99. - Charles R Greathouse IV, Mar 27 2014
MAPLE
A007497 := proc(n) options remember; if n <= 0 then RETURN(2) else numtheory[sigma](procname(n-1)); fi; end proc:
MATHEMATICA
a[1] = 2; a[n_] := a[n] = DivisorSigma[1, a[n-1]]; Table[a[n], {n, 30}]
NestList[ DivisorSigma[1, # ] &, 2, 27] (* Robert G. Wilson v, Oct 08 2010 *)
PROG
(Haskell)
a007497 n = a007497_list !! (n-1)
a007497_list = iterate a000203 2 -- Reinhard Zumkeller, Feb 27 2014
(PARI) normalize(M)={
my(P=Set(M[, 1]), f=concat(Mat(P), vector(#P))~);
for(i=1, #M~,
f[setsearch(P, M[i, 1]), 2] += M[i, 2]
);
f
};
addhelp(normalize, "normalize(M): Given a factorization matrix M, combine all like factors and order.");
sf(f)=my(v=vector(#f~, i, (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1)), g=factor(v[1])~); for(i=2, #v, g=concat(g, factor(v[i])~)); normalize(g~)
v=vector(100); v[1]=2; f=factor(2); for(i=2, #v, print1(i" "); v[i]= factorback(f=sf(f))); v \\ Charles R Greathouse IV, Mar 27 2014
(Python)
from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A007497_list = list(accumulate(repeat(2, 100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Changed the cross-reference from the tau to the sigma-function - R. J. Mathar, Feb 17 2010
STATUS
approved