A070089 -id:A070089 - cp's OEIS frontend

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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

Offset: 1

N. J. A. Sloane

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
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
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

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).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
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].
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
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]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

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
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; 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
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A070087 P(n) > P(n+1) where P(n) (A006530) is the largest prime factor of n.

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 51, 53, 55, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 76, 79, 80, 83, 86, 87, 89, 94, 95, 97, 99, 101, 103, 104, 107, 109, 111, 113, 116, 118, 119, 122, 123, 124

Offset: 1

N. J. A. Sloane, May 13 2002

nonn

Erdos conjectured that this sequence has asymptotic density 1/2.

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A006530, A070089.

f[n_] := FactorInteger[n][[ -1, 1]]; Select[ Range[125], f[ # ] > f[ # + 1] &]
With[{lpfn=Table[FactorInteger[n][[-1,1]],{n,200}]},Flatten[ Position[ Partition[ lpfn,2,1],?(#[[1]]>#[[2]]&),{1},Heads->False]]] (* _Harvey P. Dale, Sep 14 2014 *)

A071869 Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.

Original entry on oeis.org

8, 9, 20, 21, 24, 27, 32, 45, 56, 57, 77, 81, 84, 90, 91, 92, 105, 114, 120, 125, 132, 135, 140, 144, 147, 165, 168, 169, 170, 171, 175, 176, 177, 189, 200, 204, 212, 216, 220, 221, 225, 231, 234, 235, 247, 252, 260, 261, 275, 288, 289, 300, 315, 324, 345, 354

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

Erdős and Pomerance showed in 1978 that this sequence is infinite.

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Erdős and Carl Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321.

Cf. A006530, A070089, A071870, A079747, A082417-A082422.

gpf[n_] := FactorInteger[n][[-1, 1]]; ind = Position[Differences[Array[gpf, 350, 2]], ?(# > 0 &)] // Flatten; ind[[Position[Differences[ind], 1] // Flatten]] + 1 (* _Amiram Eldar, Jun 05 2022 *)

for(n=2,500,if(sign(component(component(factor(n),1),omega(n))-component(component(factor(n+1),1),omega(n+1)))+sign(component(component(factor(n+1),1),omega(n+1))-component(component(factor(n+2),1),omega(n+2)))==-2,print1(n,",")))

from sympy import factorint
A071869_list, p, q, r = [], 1, 2, 3
for n in range(2,10**4):
    p, q, r = q, r, max(factorint(n+2))
    if p < q < r:
        A071869_list.append(n) # Chai Wah Wu, Jul 24 2017

a(n) = A079747(n+1) - 1. - T. D. Noe, Nov 26 2007

A100384 a(n) = the smallest number x >= 2 such that for m = x to x + n - 1, A006530(m) increases.

Original entry on oeis.org

2, 2, 8, 8, 90, 168, 9352, 46189, 721970, 721970, 6449639, 565062156, 11336460025, 37151747513, 256994754033

Offset: 1

Labos Elemer, Dec 09 2004

A006530(m) is the largest prime factor of m.
a(16) > 3*10^11. - Donovan Johnson, Oct 24 2009
a(16) > 10^13. - Giovanni Resta, Jul 25 2013

			a(5)=90 because the largest prime factors of 90,91,92,93,94 are 5,13,23,31,47.

Cf. A006530, A070089, A071869, A100376, A100385, A358890.

from sympy import factorint
def A100384(n):
    k, a = 2, [max(factorint(m+2)) for m in range(n)]
    while True:
        for i in range(1, n):
            if a[i-1] >= a[i]:
                break
        else:
            return k
        a = a[i:] + [max(factorint(k+j+n)) for j in range(i)]
        k += i # Chai Wah Wu, Jul 24 2017

Edited by Don Reble, Jun 13 2007
a(13)-a(15) from Donovan Johnson, Oct 24 2009
Name clarified by Peter Munn, Dec 05 2022

A087429 a(n) = 1 if gpf(n) < gpf(n+1), otherwise 0, where gpf = A006530 (greatest prime factor).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Reinhard Zumkeller, Sep 02 2003

nonn

Equivalently, a(n) = 1 iff A061395(n+1) > A061395(n), otherwise a(n) = 0. - Giovanni Teofilatto, Jan 03 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Characteristic function of A070089.

Cf. A006530, A057427, A061395, A070087, A070221, A087430.

Join[{1}, Table[If[PrimePi[FactorInteger[n + 1][[ -1, 1]]] > PrimePi[FactorInteger[n][[ -1, 1]]], 1, 0], {n, 2, 90}]] (* Stefan Steinerberger, Jan 06 2008 *)
If[#[[1]]<#[[2]],1,0]&/@Partition[FactorInteger[#][[-1,1]]&/@Range[120],2,1] (* Harvey P. Dale, Nov 19 2023 *)

a(n) = A057427(1+A057427(A070221(n))).
a(p-1)=1 and a(p)=0 for primes p.
a(A070089(n)) = 1, a(A070087(n)) = 0, a(A087430(n)) = 0.

Edited by N. J. A. Sloane, Jul 01 2008, at the suggestion of R. J. Mathar

A100376 a(n) is the largest number x such that for m=n to n+x-1, A006530(m) increases.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 2, 1, 5, 4, 3, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 3

Offset: 2

Labos Elemer, Dec 09 2004

nonn

A006530(m) is the greatest prime factor (gpf) of m.

If p is an odd prime, a(p)=1, because the largest prime factor of p+1 is smaller than p.

			a(8)=4 because the largest prime factors of 8,9,10,11 are 2,3,5,11; but gpf(12)=3.
From _Michael De Vlieger_, Jul 30 2017: (Start)
Value  First position
   1         3
   2         2
   3         9
   4         8
   5        90
   6       168
   7      9352
   8     46189
   9    721971
  10    721970
(End)

Michel Marcus, Table of n, a(n) for n = 2..10000

Cf. A006530, A070089, A071869, A100384, A100387.

With[{s = Differences@ Array[FactorInteger[#][[-1, 1]] &, 115]}, Table[1 + LengthWhile[Drop[s, n], # > 0 &], {n, Length@ s - 10}]] (* Michael De Vlieger, Jul 30 2017 *)

a(n) = {m = n+1; gpf = vecmax(factor(n)[,1]); while((ngpf=vecmax(factor(m)[,1])) > gpf, m++; gpf = ngpf;); m - n;} \\ Michel Marcus, Jul 25 2017

Edited by Don Reble, Jun 13 2007

A100385 a(n) is the least number x >= 2 such that for m=x to x+n-1, A006530(m) decreases.

Original entry on oeis.org

2, 3, 13, 13, 491, 1851, 12721, 12721, 109453, 586951, 120797465, 624141002, 4044619541, 267793490438, 315400191511, 1285600699441

Offset: 1

Labos Elemer, Dec 09 2004

A006530(m) is the largest prime factor of m.
a(15) > 3*10^11. - Donovan Johnson, Oct 24 2009
a(17) > 7*10^12. - Giovanni Resta, May 04 2017

			a(5)=491 because the largest prime factors of 491,492,493,494,495 are 491,41,29,19,11.

Cf. A006530, A070087, A070089, A071870, A100384, A100386.

Function[s, Prepend[Reverse@ FoldList[If[#2 > #1, #1, #2] &, Reverse@ #], 2] &@ Map[Function[k, First@ SelectFirst[s, And[Sign@ First@ # == 1, Length@ # == k] &]], Range[Max@ Map[Length, s]]]]@ SplitBy[Flatten[ Partition[Array[{#, FactorInteger[#][[-1, 1]]} &, 10^6], 2, 1] /. {{n_, a_}, {, b}} /; n > 0 :> -n Sign[Differences@ {a, b}]], Sign] (* Michael De Vlieger, May 04 2017, Version 10.2 *)

a(n) = A070089(x)+1, where x is the smallest positive integer such that A070089(x+1)-A070089(x) >= n. - Pontus von Brömssen, Nov 09 2022

Edited by Don Reble, Jun 13 2007
a(14) from Donovan Johnson, Oct 24 2009
a(15)-a(16) from Giovanni Resta, May 04 2017

A185011 Numbers k such that P(k^2+1) < P((k+1)^2+1) where P(n) (A006530) is the largest prime factor of n.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 13, 15, 18, 19, 21, 23, 25, 27, 28, 31, 32, 34, 35, 38, 39, 41, 43, 44, 47, 48, 50, 53, 55, 57, 58, 60, 64, 65, 68, 70, 73, 75, 76, 77, 78, 80, 81, 83, 86, 87, 89, 91, 93, 96, 99, 100, 105, 107, 109, 111, 112, 114, 115, 117, 119, 123, 125

Offset: 1

Michel Lagneau, Jan 23 2012

nonn

			8 is in the sequence because 8^2+1 = 5*13 and 9^2+1 = 2*41 => 13 < 41.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002522, A070089, A006530, A070087, A185002.

f[n_]:=FactorInteger[n^2+1][[-1,1]];Select[Range[125],f[#]

A216651 Lengths of decreasing blocks of A006530, the greatest prime factor of n, starting from the second term.

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 2, 4, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 3, 4, 2, 3, 1, 2, 2, 2, 2, 2, 1, 1, 2, 4, 2, 2, 2, 2, 3, 2, 1, 3, 1, 2, 1, 3, 2, 1, 1, 1, 3, 2, 2, 2, 3, 1, 2, 2, 2, 2, 1, 2, 3, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 3, 1, 2, 1, 2, 4, 2, 4, 2, 3, 1, 2, 1

Offset: 1

Michel Lagneau, Sep 12 2012

nonn

Let gpf(m) be the greatest prime factor of m and the subset E(n) = {m, m+1, ..., m+L-1} such that gpf(m) > gpf(m+1) > ... > gpf(m+L-1) where L is the maximum length of E(n) and n the index such that {E(1) union E(2) union .... } = {2, 3, 4, ...}.
The growth of a(n) is very slow. See the following smallest values of m such that a(m) = n:
a(1) = 1, a(2) = 2, a(20) = 3, a(8) = 4, a(251) = 5, a(936) = 6, a(15553) = 7, a(6380) = 8, a(54838)=9, a(293548) = 10.

			A006530 with decreasing blocks marked: (2), (3, 2), (5, 3), (7, 2), (3), (5), (11, 3), (13, 7, 5, 2), .... Thus the terms of this sequence are 1, 2, 2, 2, 1, 1, 2, 4, ....

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530, A216650.

First differences of A070089.

N:= 1000: # to use A006530(1..N)
L:= map(max @ numtheory:-factorset, [$1..N]):
DL:= L[2..-1]-L[1..-2]:
R:= select(t -> DL[t]>= 0, [$1..N-1]):
R[2..-1]-R[1..-2]; # Robert Israel, Mar 02 2018

a(n) = A070089(n+1)-A070089(n). - Pontus von Brömssen, Nov 09 2022

A100383 Numbers k such that gpf(k) < gpf(k+1) < ... < gpf(k+9), where gpf(x) = A006530(x), the greatest prime factor of x. Numbers initiating an uphill gpf run of length 10.

Original entry on oeis.org

721970, 1091150, 6449639, 6449640, 10780550, 12161824, 15571630, 17332430, 23189750, 24901256, 28262037, 30275508, 30814114, 32184457, 32608598, 35323087, 35725704, 38265227, 38896955, 69845438, 71040720, 74345936, 79910528, 85293163, 111082114

Offset: 1

Labos Elemer, Dec 09 2004

nonn

Analogous chains of length 3 (see A071869) are infinite as shown by Erdős and Pomerance (1978). What is true for longer successions of length=4,5,...?

			n = 85293163: the corresponding uphill run of GPFs is (739, 5197, 6311, 7457, 8537, 1776941, 6561013, 8529317, 9477019, 21323293).

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321. [alternate link]

Cf. A006530, A071869, A070089, A100376, A100384.

cp's OEIS Frontend

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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A070087 P(n) > P(n+1) where P(n) (A006530) is the largest prime factor of n.

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A071869 Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.

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A100384 a(n) = the smallest number x >= 2 such that for m = x to x + n - 1, A006530(m) increases.

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A087429 a(n) = 1 if gpf(n) < gpf(n+1), otherwise 0, where gpf = A006530 (greatest prime factor).

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A100376 a(n) is the largest number x such that for m=n to n+x-1, A006530(m) increases.

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A100385 a(n) is the least number x >= 2 such that for m=x to x+n-1, A006530(m) decreases.

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