A052128 -id:A052128 - cp's OEIS frontend

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

David W. Wilson

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
a(n) = cardinality of smallest set on which idempotence of order n+1 (f^{n+1} = f) differs from idempotence of order e for 2 <= e <= n (see von Eitzen link for proof); derivable from A245501. - Mark Bowron, May 22 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Hagen von Eitzen, Least k to distinguish f^n = f on {1,...,k} from f^e = f for every 1 < e < n is largest prime power factor of n-1?, Math Stack Exchange.

Cf. A006530, A010055, A020639, A027750, A034684, A028233, A051283, A052128, A053585, A057110, A060818, A038610, A081805, A088387, A088388, A100574, A210208, A245501, A284600, A308752.

Numbers n ordered by a(n): A305325.

a034699 = last . a210208_row
-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012

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 *)

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

A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d)))); \\ Antti Karttunen, Aug 06 2018

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

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

A284600 a(n) = n/(largest prime power dividing n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 12, 1, 2, 7, 1, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 12, 5, 2, 3, 8, 1, 10

Offset: 1

Ilya Gutkovskiy, Mar 30 2017

nonn

a(n) = smallest positive number k such that n/k is a prime power.

			a(12) = 3 because 12 = 2^2*3 therefore 12/(largest prime power dividing 12) = 12/4 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example

Cf. A000961, A003557, A007913, A034699, A052126, A121289, A278568.

Has same beginning as A052128 and A114536 but is strictly different from those two sequences.

f:= n ->  n /max(map(t -> t[1]^t[2], ifactors(n)[2])):
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Apr 09 2017

Join[{1}, Table[n/Last[Select[Divisors[n], PrimePowerQ[#1] &]], {n, 2, 90}]]

from sympy import lcm
def a003418(n): return 1 if n<1 else lcm(range(1, n + 1))
def a(n):
    m=1
    while True:
        if a003418(m)%n==0: return m
        else: m+=1
print([n//a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

a(n) = n/A034699(n).
a(n) = 1 if n is a prime power (A000961).
a(n) = 2 if n is a twice odd prime power (A278568).

A076388 a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 5, 2, 15, 16, 7, 18, 1, 4, 9, 22, 5, 24, 11, 26, 3, 28, 1, 30, 31, 8, 15, 2, 5, 36, 17, 10, 3, 40, 1, 42, 7, 4, 21, 46, 13, 48, 23, 14, 9, 52, 25, 6, 1, 16, 27, 58, 7, 60, 29, 2, 63, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 22, 15, 4, 7, 78, 11, 80, 39

Offset: 1

T. D. Noe, Oct 11 2002

If n is a prime or a power of a prime, a(n) = n-1. Similar to A056737, which does not have the gcd(x,y)=1 condition.

			a(12) = 1 because of the possible (x,y) pairs, (1,12), (2,6), (3,4), the pair (3,4) yields the minimum difference and satisfies gcd(x,y)=1.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (terms 2..10000 from Ivan Neretin)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034699, A052128, A056737, A354933.

Differs from |A354988(n)| for the first time at n=60, where a(60) = 7, while A354988(60) = -11.

nMax = 100; Table[dvs = Divisors[n]; i = 1; j = 1; While[n/dvs[[i]] > dvs[[i]], If[GCD[n/dvs[[i]], dvs[[i]]] == 1, j = i]; i++ ]; n/dvs[[j]] - dvs[[j]], {n, 2, nMax}]

A076388(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(d-(n/d)))); \\ Antti Karttunen, Jun 16 2022

a(n) = A354933(n) - A052128(n). - Corrected by Antti Karttunen, Jun 16 2022

Definition formally changed from x < y to x <= y, to accommodate the prepended term a(1)=0 - Antti Karttunen, Jun 16 2022

A354933 a(1) = 1; for n > 1, a(n) = n / the largest divisor of n that is coprime to a larger divisor of n.

Original entry on oeis.org

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, 6, 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, 12, 61, 31, 9, 64, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 81, 41, 83, 12, 17, 43

Offset: 1

Antti Karttunen, Jun 16 2022

nonn

a(n) is the smallest divisor d of n such that d >= sqrt(n) and that gcd(d,n/d) = 1. For the proof see A052128. - Jianing Song, Sep 28 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A052128, A076388.

Differs from A346596 for the first time at n=60, where a(60) = 12, while A346596(60) = 15.

a[n_] := SelectFirst[Divisors[n], # >= n/# && CoprimeQ[#, n/#] &]; Array[a, 100] (* Amiram Eldar, Jun 16 2022 *)

A354933(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(d)));

a(n) = n / A052128(n).

a(n) = A052128(n) + A076388(n).

Definition rewritten by Jianing Song, Sep 28 2022

A081805 a(n) = n minus (largest prime power in n factorization); a(1) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 5, 0, 8, 0, 7, 10, 0, 0, 9, 0, 15, 14, 11, 0, 16, 0, 13, 0, 21, 0, 25, 0, 0, 22, 17, 28, 27, 0, 19, 26, 32, 0, 35, 0, 33, 36, 23, 0, 32, 0, 25, 34, 39, 0, 27, 44, 48, 38, 29, 0, 55, 0, 31, 54, 0, 52, 55, 0, 51, 46, 63, 0, 63, 0, 37, 50, 57, 66, 65, 0, 64, 0, 41, 0

Offset: 1

Benoit Cloitre, Apr 10 2003

nonn

a(n) = 0 when n is a prime power (A000961). - Michel Marcus, Dec 03 2013

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034699, A052128, A284600.

f[n_] := (n - (#[[1]]^#[[2]]) & /@ {FactorInteger[n][[-1]] })[[1]]; Array[f, 80] (* Robert G. Wilson v, Aug 07 2018 *)

A081805(n) = if(1==n,0,my(f = factor(n)); n - vecmax(vector(#f~, k, f[k, 1]^f[k, 2]))); \\ Michel Marcus Jul 24 2017 & Antti Karttunen, Aug 06 2018

a(n) = n - A034699(n). - Michel Marcus, Jul 24 2017

Term a(1) = 0 prepended by Antti Karttunen, Aug 06 2018

cp's OEIS Frontend

A034699 Largest prime power factor of n.

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A284600 a(n) = n/(largest prime power dividing n).

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A076388 a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.

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A354933 a(1) = 1; for n > 1, a(n) = n / the largest divisor of n that is coprime to a larger divisor of n.

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A081805 a(n) = n minus (largest prime power in n factorization); a(1) = 0.

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