A060266 -id:A060266 - cp's OEIS frontend

A060308 Largest prime <= 2n.

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

Offset: 1

Labos Elemer, Mar 27 2001

a(n) is the smallest k such that C(2n,n) divides k!. - Benoit Cloitre, May 30 2002
a(n) is largest prime factor of C(2n,n) = (2n)!/(n!)^2. - Alexander Adamchuk, Jul 11 2006
a(n) is also the largest prime in the interval [n,2n]. - Peter Luschny, Mar 04 2011
Odd prime p repeats (q-p)/2 times, where q > p is the next prime. In particular, every lesser of twin primes (A001359) occurs 1 time,  every lesser more than 3 of cousin primes (A023200) occurs 2 times, etc. - Vladimir Shevelev, Mar 12 2012

			n=1, 2n=2, p(1) = 2 = a(1) is the largest prime not exceeding 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Apart from initial term, same as A060265.
Cf. A007917 (largest prime <= n), A005843 (2n).
Cf. A020482, A049653, A060266, A060267, A060268, A060264.

a060308 = a007917 . a005843 -- Reinhard Zumkeller, May 25 2013

[NthPrime(#PrimesUpTo(2*n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

seq (prevprime(2*i+1), i=1..256);
seq(max(op(select(isprime,[$n..2*n]))),n=1..66); # Peter Luschny, Mar 04 2011

Table[Max[FactorInteger[(2n)!/(n!)^2]],{n,1,100}] (* Alexander Adamchuk, Jul 11 2006 *)
NextPrime[2*Range[80]+1,-1] (* Harvey P. Dale, Apr 23 2017 *)

a(n)=precprime(2*n) \\ Charles R Greathouse IV, May 24 2013

a(n) = Max[FactorInteger[(2n)!/(n!)^2]]. - Alexander Adamchuk, Jul 11 2006
a(n) = A006530(A000142(2*n)) and a(n) = A006530(A056040(2*n)). - Peter Luschny, Mar 04 2011
a(n) ~ 2*n as n tends to infinity. - Vladimir Shevelev, Mar 12 2012
a(n) = A007917(A005843(n)) = A226078(n, A067434(n)). - Reinhard Zumkeller, May 25 2013

More terms from Alexander Adamchuk, Jul 11 2006

A118750 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 3*k. a(n) = product[k=1..n] A118749(k).

Original entry on oeis.org

3, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 2565568005, 74401472145, 2306445636495, 71499814731345, 2645493145059765, 108465218947450365, 4664004414740365695, 219208207492797187665, 10302785752161467820255

Offset: 1

Jonathan Vos Post, Apr 29 2006

Differs from (after first term) A048599 "Partial products of the sequence (A001097) of twin primes" after 8th term. Differs from (after first term) A070826 "One half of product of first n primes A000040" after 9th term. Analogous to A118455 a(1)=1. a(n) = product{k=1..n} P(k), where P(k) is the largest prime <= k.

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748.

A060268 Distance of 2n from the closest prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 5, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 5, 3, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			n=13, 2n=26 surrounded by 23 and 29 which are from 26 in equal distance of 3 and min{3,3}=3=a(13).

Cf. A020482, A049653, A051702, A059959, A060266.

with(numtheory): [seq(min(nextprime(2*i)-2*i, 2*i-prevprime(2*i)), i=2...256)];

a[n_] := Min[NextPrime[2*n] - 2*n, 2*n - NextPrime[2*n, -1]]; Array[a, 100, 2] (* Amiram Eldar, Sep 16 2020 *)

a(n) = min(2*n - precprime(2*n-1), nextprime(2*n+1) - 2*n); \\ Michel Marcus, Sep 16 2020

a(n) = min(A049653(n), A060266(n)). - Michel Marcus, Sep 16 2020

A060267 Difference between 2 closest primes surrounding 2n.

Original entry on oeis.org

2, 2, 4, 4, 2, 4, 4, 2, 4, 4, 6, 6, 6, 2, 6, 6, 6, 4, 4, 2, 4, 4, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 4, 4, 2, 6, 6, 6, 4, 4, 6, 6, 6, 8, 8, 8, 8, 4, 4, 2, 4, 4, 2, 4, 4, 14, 14, 14, 14, 14, 14, 14, 4, 4, 6, 6, 6, 2, 10, 10, 10, 10, 10, 2, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 2, 10, 10, 10, 10, 10, 2, 4

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			a(3) = 2 because the closest primes to 2*3 = 6 are (5,7) and the difference between these is 2. - _Michael De Vlieger_, Nov 02 2017

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A020482, A049653, A060264, A060265, A060266, A060268.

with(numtheory): [seq(nextprime(2*i)-prevprime(2*i),i=2..256)];

Array[Subtract @@ NextPrime[#, {1, -1}] &[2 #] &, 96, 2] (* Michael De Vlieger, Nov 02 2017 *)
NextPrime[#]-NextPrime[#,-1]&/@(2*Range[2,100]) (* Harvey P. Dale, Nov 07 2017 *)

a(n) = nextprime(2*n+1) - precprime(2*n-1); \\ Michel Marcus, Sep 16 2020

A118747 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 2*k. a(n) = product[k=1..n] A060308(k).

Original entry on oeis.org

2, 6, 30, 210, 1470, 16170, 210210, 2732730, 46456410, 882671790, 16770764010, 385727572230, 8871734161290, 204049885709670, 5917446685580430, 183440847252993330, 5686666264842793230, 176286654210126590130

Offset: 1

Jonathan Vos Post, Apr 29 2006

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748.

A118752 a(n) = product[k=0..n] P(k), where P(k) is the smallest prime > 3*n. a(n) = product[k=0..n] A118751(k).

Original entry on oeis.org

2, 10, 70, 770, 10010, 170170, 3233230, 74364290, 2156564410, 62540367890, 1938751404590, 71733801969830, 2654150672883710, 108820177588232110, 4679267636293980730, 219925578905817094310, 11656055682008305998430

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A118456 a(n) = product{k=1..n} P(k), where P(k) is the smallest prime >= k.

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

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748-51.

Rest[FoldList[Times,1,Table[NextPrime[3n],{n,0,20}]]] (* Harvey P. Dale, Mar 09 2014 *)

Definition corrected by Harvey P. Dale, Mar 09 2014

A073310 a(n) is the smallest number k such that 2+k and 2n+k are both prime.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 5, 3, 1, 1, 5, 3, 1, 3, 1, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 5, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 5, 3, 1, 11, 5, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 17, 11, 9, 11, 5, 3, 1, 3, 1, 5, 3, 1, 1, 9, 9, 5, 3, 1, 1, 5, 3, 1, 5, 3, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 9, 9, 5, 3, 1, 1, 3, 1, 1, 11, 9, 29

Offset: 1

T. D. Noe, Aug 02 2002

Conjecture: a(n) < 2n. See A073316 for a generalization for all positive even numbers less than 2n.

			a(45) = 11 because 11 is the smallest number yielding two primes when added to 2 and 90. This is the first instance where this sequence differs from A060266.

Cf. A060266, A073316.

maxN=200; lst={}; For[n=1, n<=maxN, n++, k=1; While[k<2n&&!(PrimeQ[k+2]&&PrimeQ[k+2n]), k=k+2]; AppendTo[lst, k]; If[k>2n, Print["Failure at n = ", n]]]; lst

cp's OEIS Frontend

A060308 Largest prime <= 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A118750 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 3*k. a(n) = product[k=1..n] A118749(k).

Views

Author

Keywords

Comments

Crossrefs

A060268 Distance of 2n from the closest prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A060267 Difference between 2 closest primes surrounding 2n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A118747 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 2*k. a(n) = product[k=1..n] A060308(k).

Views

Author

Keywords

Crossrefs

A118752 a(n) = product[k=0..n] P(k), where P(k) is the smallest prime > 3*n. a(n) = product[k=0..n] A118751(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A073310 a(n) is the smallest number k such that 2+k and 2n+k are both prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica