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A007497
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a(1) = 2, a(n) = sigma(a(n-1)).
(Formerly M0581)
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20
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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
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OFFSET
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1,1
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COMMENTS
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Note that a(32) = 125038913126400 = 11182080^2. - Zak Seidov, Aug 29 2012
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Conjecture: (1/2)*log(n) < a(n+1)/a(n) < 2*log(n). - Benoit Cloitre, May 08 2003
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
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MAPLE
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A007497 := proc(n) options remember; if n <= 0 then RETURN(2) else numtheory[sigma](procname(n-1)); fi; end proc:
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MATHEMATICA
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a[1] = 2; a[n_] := a[n] = DivisorSigma[1, a[n-1]]; Table[a[n], {n, 30}]
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PROG
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(Haskell)
a007497 n = a007497_list !! (n-1)
(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)))
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CROSSREFS
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See also the similarly defined A051572 which has a(1) = 5 instead.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Changed the cross-reference from the tau to the sigma-function - R. J. Mathar, Feb 17 2010
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STATUS
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approved
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