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A272930 a(n) is the least k such that sigma(sigma(k)) = n*k, where sigma(n) is the sum of the divisors of n, or 0 if no such k exists.

%I A272930 #98 Jan 23 2019 16:18:03
%S A272930 1,2,8,15
%N A272930 a(n) is the least k such that sigma(sigma(k)) = n*k, where sigma(n) is the sum of the divisors of n, or 0 if no such k exists.
%C A272930 If a(5) is not zero, it exceeds 5*10^11 (see A098223). Likewise for a(17).
%C A272930 a(6) to a(16) are 42, 24, 60, 168, 480, 4404480, 2200380, 57120, 217728, 1058148, 7526400. a(18) is 39352320.
%C A272930 Is a(n) in fact nonzero for every positive n? - _Franklin T. Adams-Watters_, Jan 22 2019 [who previously conjectured that it is]
%C A272930 a(19) to a(26) are 312792480, 1505806848, 341543854080, 83825280, 13460388480, 8530704000, 58350015360, 284430182400. - _Michel Marcus_, May 18 2016
%C A272930 From _Michel Marcus_, May 18 2016; Jul 19 2016, Aug 23 2016, Sep 06 2016: (Start)
%C A272930 a(17) <= 336421458837032140800;
%C A272930 a(27) <= 4641476998878720;
%C A272930 a(28) <= 23479734980782080;
%C A272930 a(29) <= 4670834235654671884800;
%C A272930 a(30) <= 7526652811748265000960;
%C A272930 a(31) <= 45781120625942782080;
%C A272930 a(32) <= 242094947364010540800;
%C A272930 a(33) <= 216462850095065333760000;
%C A272930 a(34) <= 2366077977040955880819916800;
%C A272930 a(35) <= 8076837429313362044375040000;
%C A272930 a(36) <= 2634106558176405916291008921600;
%C A272930 a(37) <= 299500004890186577026355605378405509365760000000;
%C A272930 a(38) <= 45103591381041833364829469933568000. (End)
%H A272930 See the links in A019278. - _Altug Alkan_, May 31 2016 and May 18 2016
%e A272930 sigma(8) = 15. sigma(15) = 24 = 3*8. Since this does not work for any value smaller than 8, a(3) = 8.
%p A272930 with(numtheory):
%p A272930 a:=proc(n) local k :
%p A272930 for k while sigma(sigma(k))<>n*k do od : k end: # _Robert FERREOL_, Apr 11 2018
%t A272930 Table[SelectFirst[Range[10^2], Nest[DivisorSigma[1, #] &, #, 2] == n # &], {n, 4}] (* _Michael De Vlieger_, May 11 2016, Version 10 *)
%o A272930 (PARI) a(n)=my(r=1);while(sigma(sigma(r))!=n*r,r++);r \\ works only if a(n) is not zero.
%Y A272930 Cf. A000203 (sigma), A098223, A051027, A019279, A007539, A019278.
%K A272930 nonn,more,bref
%O A272930 1,2
%A A272930 _Franklin T. Adams-Watters_, May 11 2016