A060265 -id:A060265 - cp's OEIS frontend

A224911 Greatest prime dividing A190339(n).

2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113, 113, 113, 113, 127, 127, 131, 131

Offset: 0

Paul Curtz, Apr 19 2013

nonn

It appears that a(n) = A060308(n+1), verified for n <=420. - R. J. Mathar, Apr 28 2013
This appears to be a sequence of nondecreasing primes containing each prime at least once.
We might also consider a sequence b(n) defined by 2 followed by A006094(n): 2, 6, 15, 35, 77, 143, 221, ... . A190339(n) is also divisible by a stuttered version of b(n), namely by the sequence 2, 6, 15, 35, 35, 77, 143, 143, ... .

			a(0) = 6/2 = 3, a(1) = 15/3 = 5, a(2) = 105/15 = 7, a(3) = 105/15 = 7, a(4) = 231/21 = 11.

Cf. A006530, A060308, A065091, A190339, A120303, A060265.

A224911 := proc(n)
    A006530(A190339(n)) ;
end proc: # R. J. Mathar, Apr 25 2013

nmax = 67; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; FactorInteger[#][[-1, 1]]& /@ Denominator[Diagonal[diff]] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = A006530(A190339(n)).

A255427 a(n) = product of distinct terms of row n in triangle A255313.

Original entry on oeis.org

3, 15, 105, 35, 385, 5005, 15015, 255255, 4849845, 1616615, 37182145, 37182145, 22309287, 646969323, 20056049013, 33426748355, 33426748355, 1236789689135, 1236789689135, 50708377254535, 2180460221945005, 2180460221945005, 102481630431415235

Offset: 1

Reinhard Zumkeller, Feb 22 2015

Product of row n of table A255316;
A001221(a(n)) = A001222(a(n)) = A255395(n);
A006530(a(n)) = A060265(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A255313, A255316, A001221, A001222, A255395.

import Data.List (nub)
a255427 = product . nub . a255313_row

A055377 a(n) = largest prime <= n/2.

Original entry on oeis.org

2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37

Offset: 4

Labos Elemer, Jun 22 2000; David W. Wilson, Jun 10 2005

nonn

Also largest prime factor of any composite <= n. E.g., a(15) = 7 since 7 is the largest prime factor of {4,6,8,9,10,12,14,15}, the composites <= 15.
Also largest prime dividing A025527(n) = n!/lcm[1,...,n]. [Comment from Ray Chandler, Apr 26 2007: Primes > n/2 don't appear as factors of A025527(n) since they appear once in n! and again in the denominator lcm[1,...,n]. Primes <= n/2 appear more times in the numerator than the denominator so they appear in the fraction.]
a(n) is the largest prime factor whose exponent in the factorization of n! is greater than 1. - Michel Marcus, Nov 11 2018

			n = 10, n! = 3628800, lcm[1,...,10] = 2520, A025527(10) = 1440 = 32*9*5 so a(7) = 5 (offset = 3).

Michael De Vlieger, Table of n, a(n) for n = 4..10000

Cf. A000040, A000142, A003418, A006530, A007917, A025527, A060265, A079952, A079953.

Table[Prime@ PrimePi[n/2], {n, 4, 78}] (* Michael De Vlieger, Sep 21 2017 *)

a(n) = precprime(n/2); \\ Michel Marcus, Sep 20 2017

a(n) = Max(gpf((n+2) mod k): 1 < k < (n+2) and k not prime), with gpf=A006530 (greatest prime factor). - Reinhard Zumkeller, Mar 27 2004

Where defined, that is for n > 2, a(A000040(n)) = A000040(A079952(n)). - Peter Munn, Sep 18 2017

More terms from James Sellers, Jul 04 2000

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007

A103359 T(n,k) = Max{p: prime p divides m where n-k<=m<=n+k}, triangle read by rows: 0<=k

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 2, 5, 5, 7, 5, 5, 7, 7, 7, 3, 7, 7, 7, 7, 11, 7, 7, 7, 7, 11, 11, 13, 2, 7, 7, 11, 11, 13, 13, 13, 3, 5, 11, 11, 13, 13, 13, 13, 17, 5, 11, 11, 13, 13, 13, 13, 17, 17, 19, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 3, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 13, 13

Offset: 1

Reinhard Zumkeller, Mar 20 2005

T(n,0) = A006530(n); T(n,1) = A093074(n) for n>1;

T(n,n-1) = A060265(n) for n>1.

			Triangle begins:
1;
2, 3;
3, 3, 5;
2, 5, 5, 7;
5, 5, 7, 7, 7;
...

T[n_, k_] := FactorInteger[#][[-1, 1]]& /@ Range[n-k, n+k] // Max;
Table[T[n, k], {n, 1, 13}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

A308754 a(0) = 0, a(n) = a(n-1) + 1 if 2*n + 3 is prime, otherwise a(n) = a(n-1).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 7, 8, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 28, 28, 28, 28, 28, 28, 29, 29, 30, 30, 30, 31

Offset: 0

Carlos Fernandez Rivero, Jun 22 2019

It appears that A000040(a(n)) ~ 2*n as n tends to infinity. (See Mar 12 2012 note from Vladimir Shevelev in A060308.)

			a(0) = 0 (by definition).
a(1) = 1 = a(0) + 1, because 2*1 + 3 is prime;
a(2) = 2 = a(1) + 1, because 2*2 + 3 is prime;
a(3) = 2 = a(2),     because 2*3 + 3 is not prime;
a(4) = 3 = a(3) + 1, because 2*4 + 3 is prime.

Carlos Fernandez Rivero, Table of n, a(n) for n = 0..10000
Carlos Fernandez Rivero, Plots of a(n) for n = 0..200 and n = 0..10000

Cf. A000040, A000720, A060265, A060308, A085090, A099801, A101264.

' p(n) contains the prime sequence except for 2. p(0)=3
' output in the a(n) sequence for 0 <= n <= maxterm
ip = -1
For n = 0 To maxterm
   If (2 * n + 3) = p(ip+1) Then
      ip = ip + 1
   End If
   a(n) = ip
Next n

[#PrimesUpTo(2*n + 4) - 2: n in [0..80] ]; // Vincenzo Librandi, Aug 01 2019

a[0] = 0; a[n_] := a[n] = a[n - 1] + Boole@PrimeQ[2 n + 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 06 2019 *)

a(n) = a(n-1) + A101264(n+1), n > 0.
a(n) = A000720(2 * (n+2)) - 2.
a(n) = A099801(n+1) - 2.
a(n) = n - A210469(n+2).
A000040(a(n) + 2) = A060265(n+2).
A000040(a(n) + 2) = A060308(n+2).
A000040(a(n) + 2) = A085090(n+2), if 2*n + 3 is prime, otherwise 0.

cp's OEIS Frontend

A224911 Greatest prime dividing A190339(n).

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A255427 a(n) = product of distinct terms of row n in triangle A255313.

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A055377 a(n) = largest prime <= n/2.

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A103359 T(n,k) = Max{p: prime p divides m where n-k<=m<=n+k}, triangle read by rows: 0<=k

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A308754 a(0) = 0, a(n) = a(n-1) + 1 if 2*n + 3 is prime, otherwise a(n) = a(n-1).

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