A237428 Numbers k with following property: List all proper divisors of k. Replace any composite number in the list with its proper divisors. Repeat. Sum of remaining numbers (1's and primes) is equal to k.

6, 126, 3808, 19360, 104320, 4317184, 126764640, 1920554064, 3710312448

Offset: 1

Author

Lechoslaw Ratajczak, Feb 07 2014

Keywords

• nonn
• more

Comments

Is there a largest term? Is there any odd term?

a(8) if it exists is greater than 10^9. - Giovanni Resta, Feb 07 2014

Composite numbers k such that k = A074206(k) + Sum_{p|k} (p-1)*A074206(k/p). - Charlie Neder, Jun 02 2019

Examples

			6 is a term because: 1 + 2 + 3 = 6.
126 is a term because: [1 + 2 + 3 + (6 - 6) + 7 + (9 - 9) + (14 - 14) + (18 - 18) + (21 - 21) + (42 - 42) + (63 - 63)] + [1 + 2 + 3] + [1 + 3] + [1 + 2 + 7] + [1 + 2 + 3 + (6 - 6) + (9 - 9)] + [1 + 3 + 7] + [1 + 2 + 3 + (6 - 6) + 7 + (14 - 14) + (21 - 21)] + [1 + 3 + 7 + (9 - 9) + (21 - 21)] + [1 + 2 + 3] + [1 + 3] + [1 + 2 + 3] + [1 + 2 + 7] + [1 + 3 + 7] + [1 + 3] + [1 + 3 + 7] = 126.
		

Crossrefs

Cf. A000203, A000396, A001065, A002033, A032741, A074206, A191150.

Programs

Mathematica

v[n_] := If[PrimeQ@n, 1, Block[{s = Sum[If[e == 1 || PrimeQ@e, e, v@e], {e, Most@ Divisors@n}]}, If[n < 1000, v[n] = s, s]]]; Select[Range@ 20000, # == v@# &] (* Giovanni Resta, Feb 07 2014 *)

Extensions

a(6)-a(7) from Giovanni Resta, Feb 07 2014

a(8)-a(9) from Amiram Eldar, Jun 28 2025