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A007497 a(1) = 2, a(n) = sigma(a(n-1)).
(Formerly M0581)
19
2, 3, 4, 7, 8, 15, 24, 60, 168, 480, 1512, 4800, 15748, 28672, 65528, 122880, 393192, 1098240, 4124736, 15605760, 50328576, 149873152, 371226240, 1710858240, 7926750720, 33463001088, 109760857440, 384120963072, 1468475386560, 7157589626880, 33151875434496 (list; graph; refs; listen; history; text; internal format)
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

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..1500 (first 200 terms from 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.

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 sigma(A007497(n-1)); fi; end;

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

Cf. A000203, A175877 (positions of odd terms), A175878 (odd terms).

See also the similarly defined A051572 which has a(1) = 5 instead.

See also A257348.

Sequence in context: A092063 A227007 A126850 * A126882 A239973 A281782

Adjacent sequences:  A007494 A007495 A007496 * A007498 A007499 A007500

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

EXTENSIONS

Changed the cross-reference from the tau to the sigma-function - R. J. Mathar, Feb 17 2010

STATUS

approved

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Last modified August 16 19:22 EDT 2017. Contains 290626 sequences.