

A233511


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


2



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
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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



