A233511 Replace the largest prime factor p>2 in n (if any) with the prime preceding p.

1, 2, 2, 4, 3, 4, 5, 8, 6, 6, 7, 8, 11, 10, 9, 16, 13, 12, 17, 12, 15, 14, 19, 16, 15, 22, 18, 20, 23, 18, 29, 32, 21, 26, 25, 24, 31, 34, 33, 24, 37, 30, 41, 28, 27, 38, 43, 32, 35, 30, 39, 44, 47, 36, 35, 40, 51, 46, 53, 36, 59, 58, 45, 64, 55, 42, 61, 52, 57, 50, 67, 48, 71, 62, 45

Offset

1,2

Comments

This endomorphism a:N->N replaces the largest prime factor in n with the prime preceding it. For coherence, when there is no prime divisor or when the largest one is 2, a(n)=n. Some interesting properties: a(n)<=n; bigomega(a(n)) = bigomega(n); invariant elements of a(n) are the powers of 2 (A000079), all primes form a simple orbit terminating with 2 and containing no composite, 2^m terminates orbits of all numbers with m prime factors (with multiplicity); etc.

Links

Stanislav Sykora, Table of n, a(n) for n = 1..10000

S. Sykora, PARI scripts: PrimesRelatedFunctions

Example

a(28)=a(2*2*7)=2*2*5=20, a(20)=12, a(12)=8, a(8)=8.

Mathematica

Table[Times @@ If[Last@ # > 2, ReplacePart[#, {-1} -> NextPrime[Last@ #, -1]], #] &@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ n, {1}], {n, 75}] (* Michael De Vlieger, Apr 11 2016 *)

Prog

(PARI) A233511(n)=local(p); p=LargestPrimeFactor(n); return
((n\p)*PreviousPrime(p)) \\ See the links for the auxiliary scripts

Crossrefs

Cf. A000040 (primes), A000079 (powers of 2), A233570.

Sequence in context: A246796 A177235 A079707 * A205793 A178431 A233574

Adjacent sequences: A233508 A233509 A233510 * A233512 A233513 A233514

Keyword

nonn

Author

Stanislav Sykora, Dec 11 2013

Status

approved