A007497 a(1) = 2, a(n) = sigma(a(n-1)).
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
Offset: 1
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.
Crossrefs
Programs
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Haskell
a007497 n = a007497_list !! (n-1) a007497_list = iterate a000203 2 -- Reinhard Zumkeller, Feb 27 2014
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Maple
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
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 *) -
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
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
Extensions
Changed the cross-reference from the tau to the sigma-function - R. J. Mathar, Feb 17 2010
Comments