A060308 -id:A060308 - cp's OEIS frontend

A238691 a(n) = A190339(n)/A224911(n).

1, 2, 3, 15, 15, 21, 1155, 165, 2145, 51051, 255255, 440895, 440895, 969, 111435, 248834355, 248834355, 2927463, 5898837945, 44352165, 1641030105, 8563193457, 42815967285, 80047243185, 1360803134145, 32898537309, 7731156267615, 1028243783592795, 1028243783592795, 375840831244263

Offset: 0

Paul Curtz, Mar 03 2014

nonn

Are non-repeated terms of A224911(n) (2,3,5,11,17,...) A124588(n+1)?
Are repeated terms of A224911(n) (7,13,19,23,31,37,...) A049591(n+1)? At that sequence, Benoit Cloitre mentions a link to the Bernoulli numbers.
Greatest primes dividing a(n): 1, 2, 3, 5, 5, 7, 11, 11, 13, 17, 17, 19, 19, 19, 23, 29, 29, 29, ... = b(n). It appears that b(n) is A224911(n) with A008578(n), ancient primes, instead of A000040(n).
Hence c(n) = 2, 6, 15, 35, ... = 2, followed by A006094(n+1).

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A060308.

nmax = 40; 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]]

a(16)-a(25) from Jean-François Alcover, Mar 03 2014

A081259 a(n) is the smallest k such that C(3n,n) divides k!.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 23, 23, 29, 31, 31, 37, 41, 43, 47, 47, 53, 53, 59, 61, 61, 67, 71, 73, 73, 79, 83, 83, 89, 89, 89, 97, 101, 103, 107, 109, 113, 113, 113, 113, 113, 127, 131, 131, 137, 139, 139, 139, 149, 151, 151, 157, 157, 163, 167, 167, 173, 173, 179, 181

Offset: 1

Benoit Cloitre, Apr 20 2003

nonn

a(n) is the largest prime < 3n. More generally the smallest k such that C(mn,n) divides k! is the largest prime < mn.

Cf. A060308, A151799.

NextPrime[3Range[70],-1] (* Harvey P. Dale, Oct 20 2011 *)

a(n)=if(n<0,0,s=1; while(k!%binomial(3*n,n)>0,k++); k)

a(n) = A151799(3n). - R. J. Mathar, Jul 18 2015

A118753 First prime after 4n. Smallest prime >= 4*n. Bisection of A060264.

Original entry on oeis.org

2, 5, 11, 13, 17, 23, 29, 29, 37, 37, 41, 47, 53, 53, 59, 61, 67, 71, 73, 79, 83, 89, 89, 97, 97, 101, 107, 109, 113, 127, 127, 127, 131, 137, 137, 149, 149, 149, 157, 157, 163, 167, 173, 173, 179, 181, 191, 191, 193, 197, 211, 211, 211, 223, 223, 223, 227, 229, 233

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A060264 First prime after 2n; A118751 First prime after 3n.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000040, A008586, A060308, A060264, A118747-56.

seq(nextprime(4*k),k=0..100); # Robert Israel, Dec 25 2017

NextPrime/@(4Range[0,60]) (* Harvey P. Dale, Nov 14 2021 *)

a(n) = min{A008586(n)+k such that k>0 and A008586(n)+k in A000040}.

A118755 Smallest prime >= 6*n.

Original entry on oeis.org

2, 7, 13, 19, 29, 31, 37, 43, 53, 59, 61, 67, 73, 79, 89, 97, 97, 103, 109, 127, 127, 127, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 199, 211, 211, 223, 223, 229, 239, 241, 251, 257, 263, 269, 271, 277, 283, 293, 307, 307

Offset: 0

Jonathan Vos Post, Apr 29 2006

Cf. A007918, A008588, A060308, A060264, A118747-A118754.

A002476 is a subsequence.

Prime[1+PrimePi[6Range[0,50]]] (* T. D. Noe, Nov 15 2006 *)
NextPrime[6*Range[0,50]] (* Harvey P. Dale, Sep 05 2015 *)

a(n) = nextprime(6*n); \\ Michel Marcus, Feb 13 2021

a(n) = A007918(A008588(n)). - Michel Marcus, Feb 13 2021

Corrected by T. D. Noe, Nov 15 2006

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.

A335046 Maximal common prime of two Goldbach partitions of 2n and 2(n+1) or zero (if common prime does not exist).

Original entry on oeis.org

0, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 19, 29, 31, 31, 0, 37, 37, 41, 43, 43, 47, 47, 43, 53, 53, 43, 59, 61, 61, 0, 67, 67, 71, 73, 73, 0, 79, 79, 83, 83, 79, 89, 89, 79, 0, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 109, 0, 113, 109, 0, 127, 127, 131, 131, 127, 137, 139, 139

Offset: 2

Ivan N. Ianakiev, May 21 2020

nonn

			4 = 2+2 and 6 = 3+3. Since those are the only available Goldbach partitions and they have no common prime, a(4/2) = a(2) = 0. 14 = 3+11 and 16 = 5+11, so a(14/2) = a(7) = 11.

Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002373, A002375, A045917, A060308, A335045.

S:= proc(n) option remember; {seq((h-> `if`(
      andmap(isprime, h), h, [])[])([n+i, n-i]), i=0..n-2)}
    end:
a:= n-> max(0, (S(n) intersect S(n+1))[]):
seq(a(n), n=2..80);  # Alois P. Heinz, Jun 20 2020

d[n_]:=Flatten[Cases[FrobeniusSolve[{1,1},2*n],{?PrimeQ}]]
e[n_]:=Intersection[d[n],d[n+1]]; f[n_]:=If[e[n]=={},0,Max[e[n]]];
f/@Range[2,100]

A357253 a(n) is the largest prime < 6*n.

Original entry on oeis.org

5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 89, 101, 107, 113, 113, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 199, 211, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 293, 293, 311, 317, 317, 317, 331, 337, 347, 353

Offset: 1

Michel Marcus, Sep 20 2022

Largest prime that can be obtained after n rolls of a fair 6-sided die.

			a(11) = 61 as the largest prime <= 6*11 = 66 is 61; each of 62, 63, 64, 65 and 66 are not prime. - _David A. Corneth_, Sep 20 2022

Noga Alon and Yaakov Malinovsky, Hitting a prime in 2.43 dice rolls (on average), arXiv:2209.07698 [math.PR], 2022.

Cf. A002476, A007528, A007917, A008588, A060308, A118749, A151799.

a:= n-> prevprime(6*n):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 20 2022

a[n_] := NextPrime[6*n, -1]; Array[a, 60] (* Amiram Eldar, Sep 20 2022 *)

PARI
```
a(n) = precprime(6*n);
```

from sympy import prevprime
def A357253(n): return prevprime(6*n) # Chai Wah Wu, Sep 20 2022

a(n) = A007917(A008588(n)).

a(n) = A151799(A008588(n)), since 6*n is never prime.

A234316 Irregular triangle T, read by rows, such that row n lists the larger parts of the Goldbach partitions of 2n (in decreasing order).

Original entry on oeis.org

2, 3, 5, 7, 5, 7, 11, 7, 13, 11, 13, 11, 17, 13, 19, 17, 11, 19, 17, 13, 23, 19, 13, 23, 17, 23, 19, 17, 29, 19, 31, 29, 23, 17, 31, 29, 23, 19, 31, 19, 37, 29, 23, 37, 31, 29, 23, 41, 37, 31, 43, 41, 29, 23, 43, 41, 37, 31, 29, 47, 43, 37, 31, 47, 41, 29, 47, 43, 41, 37, 31

Offset: 2

Wesley Ivan Hurt, Dec 23 2013

Row n has first entry A060308(n), and length A045917(n). If Goldbach's conjecture is true, then each row of the triangle contains at least 1 entry.

This is the companion irregular triangle to A184995. See the first formula. - Wolfdieter Lang, May 14 2016

			The irregular triangle T(n,i) begins:
   n | 2*n | i = 1   2   3   4   5   6 ...
  ---+-----+------------------------------
   2 |   4 |     2
   3 |   6 |     3
   4 |   8 |     5
   5 |  10 |     7   5
   6 |  12 |     7
   7 |  14 |    11   7
   8 |  16 |    13  11
   9 |  18 |    13  11
  10 |  20 |    17  13
  11 |  22 |    19  17  11
  12 |  24 |    19  17  13
  13 |  26 |    23  19  13
  14 |  28 |    23  17
  15 |  30 |    23  19  17
  16 |  32 |    29  19
  17 |  34 |    31  29  23  17
  18 |  36 |    31  29  23  19
  19 |  38 |    31  19
  20 |  40 |    37  29  23
  21 |  42 |    37  31  29  23
  22 |  44 |    41  37  31
  23 |  46 |    43  41  29  23
  24 |  48 |    43  41  37  31  29
  25 |  50 |    47  43  37  31
  26 |  52 |    47  41  29
  27 |  54 |    47  43  41  37  31
  28 |  56 |    53  43  37
  29 |  58 |    53  47  41  29
  30 |  60 |    53  47  43  41  37  31
 ... Reformatted and extended. - _Wolfdieter Lang_, May 14 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A182138, A184995.

Table[First /@ DeleteDuplicates@ Map[Sort[{#, 2 n - #}, Greater] &, Select[2 n - Prime@ Range@ PrimePi[2 n], PrimeQ]], {n, 30}] // Flatten (* Michael De Vlieger, May 15 2016 *)

for(n=2, 18, forprime(p=2, n, if(isprime(2*n-p), print1(2*n-p", ")))) \\ Ralf Stephan, Dec 26 2013

T(n,i) = 2n - A184995(n,i).

T(n,i) = n + A182138(n,i). - Ralf Stephan, Dec 26 2013

A238737 a(n) = 2*n+2 - A224911(n).

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 1, 3, 1, 3, 5, 1, 3, 5, 1, 1, 3, 5, 1, 3, 1, 1, 3, 5, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 5, 7, 9, 11, 13, 1, 3, 1, 3, 5, 1, 1, 3, 5, 7, 9

Offset: 0

Paul Curtz, Mar 04 2014

nonn

It appears that a(n+2) is successively either one 1 or a string of the odd numbers.
Conjecture: the rank of 1's is A005097(n+1). This is another link between Bernoulli numbers and primes via A190339(n).
Apparently (essentially) a duplicate of A049653. - R. J. Mathar, Mar 30 2014

			a(0)=2-2=0, a(1)=4-3=1, a(2)=6-5=1, a(3)=8-7=1, a(4)=10-7=3.

Cf. A060308, A005408.

cp's OEIS Frontend

A238691 a(n) = A190339(n)/A224911(n).

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A081259 a(n) is the smallest k such that C(3n,n) divides k!.

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A118753 First prime after 4n. Smallest prime >= 4*n. Bisection of A060264.

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A118755 Smallest prime >= 6*n.

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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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A335046 Maximal common prime of two Goldbach partitions of 2n and 2(n+1) or zero (if common prime does not exist).

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A357253 a(n) is the largest prime < 6*n.

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