A053026 Maximum power of 2 arising when A000005 is applied repeatedly to n!.

1, 2, 4, 8, 16, 8, 4, 4, 4, 16, 8, 8, 8, 8, 8, 2, 8, 4, 8, 4, 4, 4, 4, 4, 4, 4, 16, 16, 4, 16, 8, 16, 4, 4, 16, 4, 4, 2, 4, 16, 8, 16, 16, 4, 8, 8, 4, 8, 8, 16, 8, 8, 32, 32, 4, 32, 4, 4, 8, 4, 2, 32, 2, 8, 4, 8, 4, 8, 8, 8, 8, 2, 8, 8, 8, 32, 32, 8, 4, 8, 8, 4, 8, 8, 8, 8, 8, 32, 8, 8, 2, 4, 2, 4, 8

Offset

1,2

Comments

Unlike the iteration of Euler phi (A000010) or cototient (A051953) functions, here the emerging powers of 2 are not accumulated at the terminal phase of iteration sequence. Non-2-powers can be intercalated.

Links

Table of n, a(n) for n=1..95.

Formula

a(n) = 2^A380802(n). - Amiram Eldar, Feb 04 2025

Example

For n = 53, the iterations are {53!, 16174080000, 840, 32, 6, 4, 3, 2}, so a(53) = 32.
For n = 130, the iterations are {130!, 287298761874053529600, 38016, 64, 7, 2}, so a(130) = 64.
For n = 563, the iterations are {563!, 2875041108020454013464609906430286933482949481627276804096000000000, 77051520, 512, 10, 4, 3, 2}, so a(563) = 512.

Mathematica

Join[{1, 2}, Table[SelectFirst[Rest[NestWhileList[DivisorSigma[0, #]&, n!, #>2&]], IntegerQ[Log[2, #]]&], {n, 3, 100}]] (* Harvey P. Dale, Jul 02 2018 *)

Prog

(PARI) a(n) = {my(m = n!); while(1 << valuation(m, 2) != m, m = numdiv(m)); m; } \\ Amiram Eldar, Feb 04 2025

Crossrefs

Cf. A000005, A000142, A000010, A051953, A036450-A036459, A380802, A380803.

Sequence in context: A228846 A070336 A348433 * A375635 A119318 A082953

Adjacent sequences: A053023 A053024 A053025 * A053027 A053028 A053029

Keyword

nonn

Author

Labos Elemer, Feb 24 2000

Status

approved