A034699 Largest prime power factor of n.

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79

Offset

1,2

Comments

n divides lcm(1, 2, ..., a(n)).

a(n) = A210208(n,A073093(n)) = largest term of n-th row in A210208. - Reinhard Zumkeller, Mar 18 2012

a(n) = smallest m > 0 such that n divides A003418(m). - Thomas Ordowski, Nov 15 2013

a(n) = n when n is a prime power (A000961). - Michel Marcus, Dec 03 2013

Conjecture: For all n between two consecutive prime numbers, all a(n) are different. - I. V. Serov, Jun 19 2019

Disproved with between p=prime(574) = 4177 and prime(575) = 4201, a(4180) = a(4199) = 19. See A308752. - Michel Marcus, Jun 19 2019

Conjecture: For any N > 0, there exist numbers n and m, N < n < n+a(n) <= m, such that all n..m are composite and a(n) = a(m). - I. V. Serov, Jun 21 2019

Conjecture: For all n between two consecutive prime numbers, all (-1)^n*a(n) are different. Checked up to 5*10^7. - I. V. Serov, Jun 23 2019

Disproved: between p = prime(460269635) = 10120168277 and p = prime(460269636) = 10120168507 the numbers n = 10120168284 and m = 10120168498 form a pair such that (-1)^n*a(n) = (-1)^m*a(m) = 107. - L. Joris Perrenet, Jan 05 2020

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)

Formula

If n = p_1^e_1 *...* p_k^e_k, p_1 < ... < p_k primes, then a(n) = Max_i p_i^e_i.

a(n) = A088387(n)^A088388(n). - Antti Karttunen, Jul 22 2018

a(n) = n/A284600(n) = n - A081805(n) = A034684(n) + A100574(n). - Antti Karttunen, Aug 06 2018

a(n) = a(m) iff m = d*a(n), where d is a divisor of A038610(a(n)). - I. V. Serov, Jun 19 2019

Mathematica

f[n_] := If[n == 1, 1, Max[ #[[1]]^#[[2]] & /@ FactorInteger@n]]; Array[f, 79] (* Robert G. Wilson v, Sep 02 2006 *)
Array[Max[Power @@@ FactorInteger@ #] &, 79] (* Michael De Vlieger, Jul 26 2018 *)

Prog

(Haskell)
a034699 = last . a210208_row
-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012
(PARI) a(n) = if(1==n, n, my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ Charles R Greathouse IV, Nov 20 2012, check for a(1) added by Antti Karttunen, Aug 06 2018
(PARI) A034699(n) = if(1==n, n, fordiv(n, d, if(isprimepower(n/d), return(n/d)))); \\ Antti Karttunen, Aug 06 2018
(Python)
from sympy import factorint
def A034699(n): return max((p**e for p, e in factorint(n).items()), default=1) # Chai Wah Wu, Apr 17 2023

Crossrefs

Cf. A006530, A010055, A020639, A027750, A034684, A028233, A051283, A052128, A053585, A057110, A060818, A038610, A081805, A088387, A088388, A100574, A210208, A284600, A308752.

Numbers n ordered by a(n): A305325.

Sequence in context: A274346 A345993 A330691 * A354933 A346596 A324388

Adjacent sequences: A034696 A034697 A034698 * A034700 A034701 A034702

Keyword

nonn,easy,nice

Author

David W. Wilson

Status

approved