A100384 -id:A100384 - cp's OEIS frontend

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

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

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

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

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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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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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.

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