A070087 -id:A070087 - cp's OEIS frontend

A185002 Numbers k such that P(k^2+1) > P((k+1)^2+1) where P(n) (A006530) is the largest prime factor of n.

4, 6, 10, 11, 12, 14, 16, 17, 20, 22, 24, 26, 29, 30, 33, 36, 37, 40, 42, 45, 46, 49, 51, 52, 54, 56, 59, 61, 62, 63, 66, 67, 69, 71, 72, 74, 79, 82, 84, 85, 88, 90, 92, 94, 95, 97, 98, 101, 102, 103, 104, 106, 108, 110, 113, 116, 118, 120, 121, 122, 124, 126

Offset: 1

Michel Lagneau, Jan 23 2012

nonn

			11 is in the sequence because 11^2+1 = 2*61 and 12^2+1 = 5*29 => 61 > 29.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A014442, A002522, A006530, A070087, A185011.

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

r=2;for(k=1,1e3,t=factor((k+1)^2+1)[,1];t=t[#t];if(tCharles R Greathouse IV, Jan 23 2012

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[#]

A216650 Maximum length of each subsequence whose elements are the greatest prime divisors of the integers 2, 3, 4, ... in increasing order.

Original entry on oeis.org

2, 2, 2, 4, 2, 1, 1, 2, 2, 4, 3, 3, 2, 3, 1, 2, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 4, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 3, 1, 3, 3, 1, 2, 5, 1, 2, 2, 2, 2, 1, 3, 2, 2, 2, 3, 2, 1, 3, 1, 1, 3, 2, 2, 3, 3, 2, 3, 1, 3, 3, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 6, 1, 5, 2, 2, 2

Offset: 1

Michel Lagneau, Sep 12 2012

nonn

Let gpf(m) = A006530(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, ...}.
See the examples for the structure of the subsequences of increasing prime divisors.
The growth of a(n) is very slow. See the following smallest values of m such that a(m) = n:
a(6) = 1, a(1) = 2, a(11) = 3, a(4) = 4, a(44) = 5, a(82) = 6, a(4672) = 7, a(23001) = 8, a(360896) = 9.

			Subset 1: {2, 3} obtained with the numbers 2, 3 => a(1) = 2;
Subset 2: {2, 5} obtained with the numbers 4, 5 => a(2) = 2;
Subset 3: {3, 7} obtained with the numbers 6, 7 => a(3) = 2;
Subset 4: {2, 3, 5, 11} obtained with the numbers 8, 9, 10, 11 => a(4) = 4.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A006530, A070087, A216651.

with(numtheory):p0:=2:it:=1:for n from 3 to 200 do: x:=factorset(n):n1:=nops(x):p:=x[n1]:if p>p0 then it:=it+1:p0:=p:else printf(`%d, `,it):it:=1:p0:=p:fi:od:

a(n) = A070087(n)-A070087(n-1) for n >= 2. - Pontus von Brömssen, Nov 09 2022

A100387 a(n) is the largest number x such that for m=n to n+x-1, A006530(m) decreases.

Original entry on oeis.org

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

Offset: 2

Labos Elemer, Dec 10 2004

nonn

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

			a(13)=4 because the largest prime factors of 13,14,15,16 are 13,7,5,2; but A006530(17)=17.

Cf. A006530, A070087, A070089, A071870, A079748, A100376, A100385, A100386.

Mathematica
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From Pontus von Brömssen, Nov 09 2022: (Start)
a(n) = 1 if and only if n >= 2 and n is a term of A070089.
If a(n) > 1 then a(n) = a(n+1)+1.
(End)

Edited by Don Reble, Jun 13 2007

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A185002 Numbers k such that P(k^2+1) > P((k+1)^2+1) where P(n) (A006530) is the largest prime factor of n.

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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.

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A216650 Maximum length of each subsequence whose elements are the greatest prime divisors of the integers 2, 3, 4, ... in increasing order.

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A100387 a(n) is the largest number x such that for m=n to n+x-1, A006530(m) decreases.

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