|
%I M0581 #71 Oct 29 2023 21:51:34
%S 2,3,4,7,8,15,24,60,168,480,1512,4800,15748,28672,65528,122880,393192,
%T 1098240,4124736,15605760,50328576,149873152,371226240,1710858240,
%U 7926750720,33463001088,109760857440,384120963072,1468475386560,7157589626880,33151875434496
%N a(1) = 2, a(n) = sigma(a(n-1)).
%C Note that a(32) = 125038913126400 = 11182080^2. - _Zak Seidov_, Aug 29 2012
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Charles R Greathouse IV, <a href="/A007497/b007497.txt">Table of n, a(n) for n = 1..1500</a> (first 200 terms from T. D. Noe)
%H G. L. Cohen and H. J. J. te Riele, <a href="http://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 91-100.
%H Graeme L. Cohen, Herman J. J. te Riele, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.868">Iterating the Sum-of-Divisors Function</a>, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100.
%H R. G. Wilson, V, <a href="/A007497/a007497.pdf">Notes, n.d.</a>
%F Conjecture: (1/2)*log(n) < a(n+1)/a(n) < 2*log(n). - _Benoit Cloitre_, May 08 2003
%F Conjecture: a(n) == 0 mod 9 for n > 34. - _Ivan N. Ianakiev_, Mar 27 2014
%F 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
%p A007497 := proc(n) options remember; if n <= 0 then RETURN(2) else numtheory[sigma](procname(n-1)); fi; end proc:
%t a[1] = 2; a[n_] := a[n] = DivisorSigma[1, a[n-1]]; Table[a[n], {n, 30}]
%t NestList[ DivisorSigma[1, # ] &, 2, 27] (* _Robert G. Wilson v_, Oct 08 2010 *)
%o (Haskell)
%o a007497 n = a007497_list !! (n-1)
%o a007497_list = iterate a000203 2 -- _Reinhard Zumkeller_, Feb 27 2014
%o (PARI) normalize(M)={
%o my(P=Set(M[,1]),f=concat(Mat(P),vector(#P))~);
%o for(i=1,#M~,
%o f[setsearch(P,M[i,1]),2] += M[i,2]
%o );
%o f
%o };
%o addhelp(normalize, "normalize(M): Given a factorization matrix M, combine all like factors and order.");
%o 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~)
%o 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
%o (Python)
%o from itertools import accumulate, repeat # requires Python 3.2 or higher
%o from sympy import divisor_sigma
%o A007497_list = list(accumulate(repeat(2,100), lambda x, _: divisor_sigma(x)))
%o # _Chai Wah Wu_, May 02 2015
%Y Cf. A000203, A175877 (positions of odd terms), A175878 (odd terms).
%Y See also the similarly defined A051572 which has a(1) = 5 instead.
%Y See also A257348.
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E Changed the cross-reference from the tau to the sigma-function - _R. J. Mathar_, Feb 17 2010
|
