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.

1, 2, 8, 15

Offset: 1

Author

Franklin T. Adams-Watters, May 11 2016

Keywords

• nonn
• more
• bref

Comments

If a(5) is not zero, it exceeds 5*10^11 (see A098223). Likewise for a(17).

a(6) to a(16) are 42, 24, 60, 168, 480, 4404480, 2200380, 57120, 217728, 1058148, 7526400. a(18) is 39352320.

Is a(n) in fact nonzero for every positive n? - Franklin T. Adams-Watters, Jan 22 2019 [who previously conjectured that it is]

a(19) to a(26) are 312792480, 1505806848, 341543854080, 83825280, 13460388480, 8530704000, 58350015360, 284430182400. - Michel Marcus, May 18 2016

From Michel Marcus, May 18 2016; Jul 19 2016, Aug 23 2016, Sep 06 2016: (Start)

a(17) <= 336421458837032140800;

a(27) <= 4641476998878720;

a(28) <= 23479734980782080;

a(29) <= 4670834235654671884800;

a(30) <= 7526652811748265000960;

a(31) <= 45781120625942782080;

a(32) <= 242094947364010540800;

a(33) <= 216462850095065333760000;

a(34) <= 2366077977040955880819916800;

a(35) <= 8076837429313362044375040000;

a(36) <= 2634106558176405916291008921600;

a(37) <= 299500004890186577026355605378405509365760000000;

a(38) <= 45103591381041833364829469933568000. (End)

Examples

			sigma(8) = 15. sigma(15) = 24 = 3*8. Since this does not work for any value smaller than 8, a(3) = 8.
		

Links

• See the links in A019278. - _Altug Alkan_, May 31 2016 and May 18 2016

Crossrefs

Cf. A000203 (sigma), A098223, A051027, A019279, A007539, A019278.

Programs

Maple

with(numtheory):
a:=proc(n) local k :
for k while sigma(sigma(k))<>n*k do od : k end: # Robert FERREOL, Apr 11 2018

Mathematica

Table[SelectFirst[Range[10^2], Nest[DivisorSigma[1, #] &, #, 2] == n # &], {n, 4}] (* Michael De Vlieger, May 11 2016, Version 10 *)

PARI

a(n)=my(r=1);while(sigma(sigma(r))!=n*r,r++);r \\ works only if a(n) is not zero.