A361421 Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.

840, 2040, 4440, 9240, 25320, 51000, 117480, 271320, 765480, 1531320, 3721800, 5956440, 12295560, 25086840, 54141960, 108284280, 250301640, 502213560, 1007626440, 2017856760, 4039750920, 8079502200, 19596145800, 44369345400, 71495068200, 115576350360, 231152701080

Offset: 1

Author

Amiram Eldar, Mar 11 2023

Keywords

• nonn

Comments

First differs from A045477 at n = 12.

840 is the least number whose infinitary aliquot sequence is not known to be finite or eventually periodic.

R. J. Mathar found that this sequence does not reach 0 or enter a cycle before the 1500th term (see A127661). This limit was extended to beyond the 3000th term (see the b-file).

Examples

			a(1) = 840 by definition.
a(2) = A126168(a(1)) = A126168(840) = 2040.
a(3) = A126168(a(2)) = A126168(2040) = 4440.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..3132

Crossrefs

Cf. A008892, A045477, A126168, A127661, A293355.

Programs

Mathematica

f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
infs[n_] := If[n==1, 1, Times @@ f @@@ FactorInteger[n]] - n; infs[0] = 0;
seq[len_, init_] := NestWhileList[infs, init, UnsameQ, All, len];
seq[27, 840]

PARI

s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(k = 840); for(n = 1, nmax, print1(k, ", "); if(k == 0, break); k = s(k)); }