A081805 -id:A081805 - 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

A290203 Numbers m having greatest prime power divisor d such that d is smaller than the difference between m and the largest prime smaller than m.

Original entry on oeis.org

126, 210, 330, 630, 1144, 1360, 2520, 2574, 2992, 3432, 3960, 4199, 4620, 5544, 5610, 5775, 5980, 6006, 6930, 7280, 8008, 8415, 9576, 10005, 10032, 12870, 12880, 13090, 14280, 14586, 15708, 15725, 16182, 17290, 18480, 18837, 19635, 19656, 20475, 20592, 22610

Offset: 1

Sílvia Casacuberta Puig, Jul 24 2017

nonn

It is conjectured that for all integers m there exist two primes p and r such that all the binomial coefficients (m,k) with 1 <= k <= m-1 are divisible by either p or r. Using Lucas's Theorem we can prove that the conjecture is true for integers m such that the difference between m and the largest prime smaller than m is smaller than the greatest prime power divisor of m. Therefore this list examines the numbers m that do not satisfy this property.

Also numbers m such that m - A007917(m) > A034699(m). - David A. Corneth, Jul 24 2017

			The first number of the sequence is 126. The prime factorization of 126 is 2*3^2*7. Therefore, the greatest prime power divisor is 9. The largest prime smaller than 126 is 113, and then the difference between 126 and 113 is 13. Then 13 is larger than 9 and therefore 126 is part of the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Sílvia Casacuberta Puig, On the divisibility of binomial coefficients, 2018; see also, arXiv:1906.07652 [math.NT], 2019.
E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, 44:93-146, 1852.
E. Lucas, Théorie des fonctions numériques simplement périodiques, American Journal of Mathematics, 44:184-196, 1878.
J. Shareshian and R. Woodroofe, Divisibility of binomial coefficients and generation of alternating groups, arXiv:1505.05143 [math.CO], 2015-2017.
Wikipedia, Kummer's Theorem
Wikipedia, Lucas' Theorem

Cf. A007917, A034699, A081805.

Reap[For[k = 3, k < 30000, k++, If[k - NextPrime[k, -1] > Max[Power @@@ FactorInteger[k]], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jul 24 2017 *)

isok(n)=my(f = factor(n)); n - precprime(n) > vecmax(vector(#f~, k, f[k,1]^f[k,2])); \\ Michel Marcus, Jul 24 2017

list(lim)=my(v=List(),p=2,f); forfactored(n=3,lim\1, f=n[2]; if(f[,2]==[1]~, p=n[1]; next); if(n[1]-p > vecmax(vector(#f~, i, f[i,1]^f[i,2])), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 24 2017

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A034699 Largest prime power factor of n.

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A290203 Numbers m having greatest prime power divisor d such that d is smaller than the difference between m and the largest prime smaller than m.

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Mathematica

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