# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a007497 Showing 1-1 of 1 %I A007497 M0581 #73 Aug 06 2024 07:02:32 %S A007497 2,3,4,7,8,15,24,60,168,480,1512,4800,15748,28672,65528,122880,393192, %T A007497 1098240,4124736,15605760,50328576,149873152,371226240,1710858240, %U A007497 7926750720,33463001088,109760857440,384120963072,1468475386560,7157589626880,33151875434496 %N A007497 a(1) = 2, a(n) = sigma(a(n-1)). %C A007497 Note that a(32) = 125038913126400 = 11182080^2. - _Zak Seidov_, Aug 29 2012 %D A007497 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A007497 Charles R Greathouse IV, Table of n, a(n) for n = 1..1500 (first 200 terms from T. D. Noe) %H A007497 G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100. %H A007497 Graeme L. Cohen, Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100. %H A007497 R. G. Wilson, V, Notes, n.d. %F A007497 Conjecture: (1/2)*log(n) < a(n+1)/a(n) < 2*log(n). - _Benoit Cloitre_, May 08 2003 %F A007497 Conjecture: a(n) == 0 mod 9 for n > 34. - _Ivan N. Ianakiev_, Mar 27 2014 %F A007497 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 A007497 := proc(n) options remember; if n <= 0 then RETURN(2) else numtheory[sigma](procname(n-1)); fi; end proc: %t A007497 a[1] = 2; a[n_] := a[n] = DivisorSigma[1, a[n-1]]; Table[a[n], {n, 30}] %t A007497 NestList[ DivisorSigma[1, # ] &, 2, 27] (* _Robert G. Wilson v_, Oct 08 2010 *) %o A007497 (Haskell) %o A007497 a007497 n = a007497_list !! (n-1) %o A007497 a007497_list = iterate a000203 2 -- _Reinhard Zumkeller_, Feb 27 2014 %o A007497 (PARI) normalize(M)={ %o A007497 my(P=Set(M[,1]),f=concat(Mat(P),vector(#P))~); %o A007497 for(i=1,#M~, %o A007497 f[setsearch(P,M[i,1]),2] += M[i,2] %o A007497 ); %o A007497 f %o A007497 }; %o A007497 addhelp(normalize, "normalize(M): Given a factorization matrix M, combine all like factors and order."); %o A007497 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 A007497 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 A007497 (Python) %o A007497 from itertools import accumulate, repeat # requires Python 3.2 or higher %o A007497 from sympy import divisor_sigma %o A007497 A007497_list = list(accumulate(repeat(2,100), lambda x, _: divisor_sigma(x))) %o A007497 # _Chai Wah Wu_, May 02 2015 %Y A007497 Cf. A000203, A175877 (positions of odd terms), A175878 (odd terms). %Y A007497 See also the similarly defined A051572 which has a(1) = 5 instead. %Y A007497 See also A257348. %K A007497 nonn,easy,nice %O A007497 1,1 %A A007497 _N. J. A. Sloane_, _Robert G. Wilson v_ %E A007497 Changed the cross-reference from the tau to the sigma-function - _R. J. Mathar_, Feb 17 2010 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE