A070087 -id:A070087 - 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

A070089 P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 36, 40, 42, 45, 46, 48, 50, 52, 54, 56, 57, 58, 60, 64, 66, 68, 70, 72, 75, 77, 78, 81, 82, 84, 85, 88, 90, 91, 92, 93, 96, 98, 100, 102, 105, 106, 108, 110, 112, 114, 115, 117

Offset: 1

N. J. A. Sloane, May 13 2002

nonn

Erdős conjectured that this sequence has asymptotic density 1/2.
There are 500149 terms in this sequence up to 10^6, 4999951 up to 10^7, 49997566 up to 10^8, and 499992458 up to 10^9. With a binomial model with p = 1/2, these would be +0.3, -0.5, -0.0, and -0.5 standard deviations from their respective means. In other words, Erdős's conjecture seems solid. - Charles R Greathouse IV, Oct 27 2015
Erdős and Pomerance (1978) proved that the lower density of this sequence is at least 0.0099. This value was improved to 0.05544 (De La Bretèche et al., 2005), 0.1063 (Wang, 2017), 0.1356 (Wang, 2018), and 0.2017 (Lü and Wang, 2018). - Amiram Eldar, Aug 02 2020

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
Régis De La Bretèche, Carl Pomerance and Gérald Tenenbaum, Products of ratios of consecutive integers, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 131-138, alternative link.
Paul Erdős and Carl Pomerance, On the largest prime factors of n and n + 1, Aequationes Math., Vol. 17, No. 1 (1978), pp. 311-321, alternative link.
Xiaodong Lü and Zhiwei Wang, On the largest prime factors of consecutive integers, 2018.
Zhiwei Wang, On the largest prime factors of consecutive integers in short intervals, Proceedings of the American Mathematical Society, Vol. 145, No. 8 (2017), pp. 3211-3220.
Zhiwei Wang, Sur les plus grands facteurs premiers d'entiers consécutifs, Mathematika, Vol. 64, No. 2 (2018), pp. 343-379, preprint, arXiv:1706.02980 [math.NT], 2017.

Cf. A006530, A070087.

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

gpf(n)=if(n<3,n,my(f=factor(n)[,1]); f[#f])
is(n)=gpf(n) < gpf(n+1) \\ Charles R Greathouse IV, Oct 27 2015

A071870 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

13, 14, 34, 37, 38, 43, 61, 62, 73, 79, 86, 94, 103, 118, 122, 123, 142, 151, 152, 157, 158, 163, 173, 185, 193, 194, 202, 206, 214, 218, 223, 229, 241, 254, 257, 258, 271, 277, 278, 283, 284, 295, 298, 302, 313, 317, 318, 321, 322, 326, 331, 334, 341, 373

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

Erdős conjectured that this sequence is infinite.

Balog (2001) proved that this sequence is infinite. - Amiram Eldar, Aug 02 2020

			13 is a term since gpf(13) = 13, gpf(14) = 7, gpf(15) = 5, and 13 > 7 > 5.

T. D. Noe, Table of n, a(n) for n = 1..1000
Antal Balog, On triplets with descending largest prime factors, Studia Scientiarum Mathematicarum Hungarica, Vol. 38, No. 1-4 (2001), pp. 45-50.
P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321. [alternate link]
Sungjin Kim, Two Remarks on the Largest Prime Factors of n and n+1, J. of Integer Sequences, 23 (2020), #20.10.1.

Cf. A006530, A070087, A071869, A082417, A082418, A082419, A082420, A082421, A082422.

Select[ Range[400], FactorInteger[#][[-1, 1]] >  FactorInteger[# + 1][[-1, 1]] > FactorInteger[# + 2][[-1, 1]] &] (* Jean-François Alcover, Jun 17 2013 *)

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,",")))

gpf(n) = vecmax(factor(n)[,1]);
isok(k) = (gpf(k) > gpf(k+1)) && (gpf(k+1) > gpf(k+2)); \\ Michel Marcus, Nov 02 2020

from sympy import factorint
A071870_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:
        A071870_list.append(n) # Chai Wah Wu, Jul 24 2017

A103667 Primes p such that the largest prime divisor of p-1 is greater than the largest prime divisor of p+1.

Original entry on oeis.org

7, 11, 23, 29, 31, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 173, 179, 191, 199, 223, 227, 233, 239, 263, 269, 293, 307, 311, 317, 347, 349, 359, 367, 373, 383, 389, 419, 431, 439, 449, 461, 467, 479, 499, 503, 509, 557, 563, 569, 571, 587

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

Primes of the form 2*A070087(n)+1 for some n. - Charles R Greathouse IV, Dec 22 2022

Conjecture: this sequence is of positive relative density in the primes, perhaps even 1/2. - Charles R Greathouse IV, Dec 22 2022

			a(1)=7 because the largest prime divisor of 6 is greater than the largest prime divisor of 8.

Karl Hovekamp, Table of n, a(n) for n = 1..39328

Cf. A023503 (greatest prime divisor of n-th prime - 1), A023509 (greatest prime divisor of n-th prime + 1), A103666, A070087.

filter:= p -> isprime(p) and max(numtheory:-factorset(p-1)) > max(numtheory:-factorset(p+1)):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jan 15 2024

Select[Prime@Range[2, 107], If[FactorInteger[#-1][[-1, 1]]>FactorInteger[#+1][[-1, 1]], True]&] (* James C. McMahon, Jan 15 2024 *)

A358890 a(n) is the first term of the first maximal run of n consecutive numbers with increasing greatest prime factors.

Original entry on oeis.org

14, 4, 1, 8, 90, 168, 9352, 46189, 2515371, 721970, 6449639, 565062156, 11336460025, 37151747513, 256994754033, 14037913234203

Offset: 1

Reinhard Zumkeller, Jan 10 2003

a(16) > 10^13. - Giovanni Resta, Jul 25 2013
The convention gpf(1) = A006530(1) = 1 is used (otherwise we would have a(2) = 2 and a(3) = 24). - Pontus von Brömssen, Dec 05 2022
a(17) > 10^14. - Martin Ehrenstein, Dec 10 2022

			a(7) = 9352 because the first sequence of seven consecutive numbers with increasing greatest prime factors is 9352=167*7*2^3, 9353=199*47, 9354=1559*3*2, 9355=1871*5, 9356=2339*2^2, 9357=3119*3, and 9358=4679*2. [Corrected by _Jon E. Schoenfield_, Sep 21 2022]

Cf. A006530, A070087, A079748, A079749 (erroneous version), A100384.

V:= Vector(11): count:= 0:
a:= 1: m:= 1: w:= 1:
for k from 2 while count < 11 do
  v:= max(numtheory:-factorset(k));
  if v > m then m:= v
  else
    if V[k-a] = 0 then V[k-a]:= a; count:= count+1; fi;
    a:= k; m:= v;
  fi
od:
convert(V,list); # Robert Israel, Dec 05 2022

from sympy import factorint
def A358890(n):
    m = 1
    gpf1 = 1
    k = 1
    while 1:
        while 1:
            gpf2 = max(factorint(m+k))
            if gpf2 < gpf1: break
            gpf1 = gpf2
            k += 1
        if k == n: return m
        m += k
        gpf1 = gpf2
        k = 1 # Pontus von Brömssen, Dec 05 2022

A079748(a(n)) = n-1.
From Pontus von Brömssen, Dec 05 2022: (Start)
A079748(a(n)-1) = 0 for n != 3.
For n != 3, a(n) = A070087(m)+1, where m is the smallest positive integer such that A070087(m+1) - A070087(m) = n.
(End)

More terms from Don Reble, Jan 17 2003
Corrected by Jud McCranie, Feb 11 2003
a(14)-a(15) from Giovanni Resta, Jul 25 2013
Name edited, a(1) and a(2) corrected by Pontus von Brömssen, Dec 05 2022
a(16) from Martin Ehrenstein, Dec 07 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

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

A079748 Largest k such that the greatest prime factors from n to n+k are monotonically increasing.

Original entry on oeis.org

2, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 2, 1, 0, 2, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 4, 3, 2, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2

Offset: 1

Reinhard Zumkeller, Jan 10 2003

nonn

A006530(n+i) < A006530(n+j) for 0 <= i < j < a(n);

if a(n) > 0 then a(n+1) = a(n) - 1.

			n=20: 20 = 5*2^2, 21 = 7*3, 22 = 11*2 and 23, followed by 24 = 3*2^3: therefore a(20)=3 (5 < 7 < 11 < 23 and 23 > 3).

Cf. A006530, A070087, A079747, A358890.

a(n) = 0 if and only if n is a term of A070087. - Pontus von Brömssen, Nov 09 2022

A100386 Numbers n such that for m=n to n+9, A006530(m) decreases.

Original entry on oeis.org

586951, 1473257, 4982941, 13565441, 24954141, 25384714, 26576686, 32026196, 35797623, 35953989, 37972276, 39048260, 51755761, 58769257, 60682681, 71342703, 77863117, 80826231, 84766857, 89768134, 98363506, 110482826, 115045547, 115898807, 120797465

Offset: 1

Labos Elemer, Dec 09 2004

nonn

A006530(n) is the largest prime factor of n.

			586951 is here because the largest prime factors of 586951..586960 are 586951,73369,21739,9467,1319,1193,1181,1091,677,29.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A006530, A070087, A071870, A100385.

<?(Max[Differences[#]]<0&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Sep 18 2016 *)

Edited by Don Reble, Jun 13 2007

A087430 Nonprimes n with gpf(n) > gpf(n+1), where gpf=A006530 (greatest prime factor).

Original entry on oeis.org

14, 15, 26, 34, 35, 38, 39, 44, 49, 51, 55, 62, 63, 65, 69, 74, 76, 80, 86, 87, 94, 95, 99, 104, 111, 116, 118, 119, 122, 123, 124, 129, 134, 142, 143, 146, 152, 153, 155, 158, 159, 161, 164, 174, 183, 185, 186, 188, 194, 195, 202, 203, 206, 207, 209, 214, 215

Offset: 1

Reinhard Zumkeller, Sep 02 2003

nonn

Subsequence of A070087.

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

Cf. A087429.

N:= 1000: # to get all terms < N
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  V[[seq(k*p,k=1..N/p)]]:= p
od:
select(t -> V[t] > V[t+1] and not isprime(t), [$1..N-1]); # Robert Israel, Jul 03 2018

With[{nn=250},Select[Complement[Range[nn],Prime[Range[PrimePi[ nn]]]], FactorInteger[ #][[-1,1]]>FactorInteger[#+1][[-1,1]]&]] (* Harvey P. Dale, Jul 14 2016 *)

cp's OEIS Frontend

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

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A070089 P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.

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A071870 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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A103667 Primes p such that the largest prime divisor of p-1 is greater than the largest prime divisor of p+1.

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A358890 a(n) is the first term of the first maximal run of n consecutive numbers with increasing greatest prime factors.

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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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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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