|
|
A006530
|
|
Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.
(Formerly M0428)
|
|
1009
|
|
|
1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021
|
|
REFERENCES
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
|
|
FORMULA
|
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
|
|
MAPLE
|
with(numtheory, divisors); A006530 := proc(n) local i, t1, t2, t3, t4, t5; t1 := divisors(n); t2 := convert(t1, list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) A006530(n)=if(n>1, vecmax(factor(n)[, 1]), 1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015
(Magma) [ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ] // Klaus Brockhaus, Oct 23 2008
(Scheme)
;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(Python)
from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice,easy,core
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|