A049653 -id:A049653 - cp's OEIS frontend

A049711 a(n) = n - prevprime(n).

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

Offset: 3

N. J. A. Sloane

nonn

All runs end in even numbers at a(p), new highs are found at A000101 and the increasing gap size is A005250. - Robert G. Wilson v, Dec 07 2001

All terms are positive since here the variant 2 (A151799(n) < n) of the prevprime function is used, rather than the variant 1 (A007917(n) <= n). - M. F. Hasler, Sep 09 2015

Cf. A000101, A005250, A049613, A049653, A049716, A049847, A064722.
Cf. A151799, A007917.
Cf. A175851.

Maple
```
A049711 := n-> n-prevprime(n);
```

PrevPrim[n_] := Block[ {k = n - 1}, While[ !PrimeQ[k], k-- ]; Return[k]]; Table[ n - PrevPrim[n], {n, 3, 100} ]
Array[#-NextPrime[#,-1]&,100,3] (* Harvey P. Dale, Dec 07 2011 *)

A049711(n)=n-precprime(n-1) \\ M. F. Hasler, Sep 09 2015

a(n) = A064722(n-1) + 1. - Pontus von Brömssen, Jul 31 2022

A060308 Largest prime <= 2n.

Original entry on oeis.org

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

A060264 First prime after 2n.

Original entry on oeis.org

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

Offset: 0

Labos Elemer, Mar 23 2001

Conjecture: for n > 2, this is the least prime p such that 1^2, 2^2, 3^2, ..., n^2 are distinct mod p. Checked to 10^4. - Charles R Greathouse IV, Dec 03 2022

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A020482, A049653, A059786, A060308, A088633, A151800.

Cf. A005843.

a060264 = a151800 . (* 2)  -- Reinhard Zumkeller, Nov 15 2013

A060264 := proc(n)
        nextprime(2*n) ;
end proc:

Table[NextPrime[2n],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

a(n) = nextprime(2*n + 1)  \\ Harry J. Smith, Jul 03 2009

a(n) = A151800(2*n). - Reinhard Zumkeller, Nov 15 2013

A060265 Largest prime less than 2n.

Original entry on oeis.org

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: 2

Labos Elemer, Mar 23 2001

a(n) = A007917(2*n) = A255313(n-1,1) = A255316(n-1,1) = A006530(A255427(n)). - Reinhard Zumkeller, Feb 22 2015

Harry J. Smith, Table of n, a(n) for n = 2..1000

Apart from initial term, same as A060308.

Cf. A020482, A049653, A088631, A007917, A005843, A255313.

a060265 = a007917 . (* 2)  -- Reinhard Zumkeller, Feb 22 2015

Maple
```
seq (prevprime(2*i+1), i=2..256);
```

Table[NextPrime[2 n, -1], {n, 2, 66}] (* Michael De Vlieger, Jul 04 2016 *)

a(n) = precprime(2*n-1) \\ Harry J. Smith, Jul 03 2009

A060266 Difference between 2n and the following 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, 7, 5, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 13, 11, 9, 7, 5, 3, 1, 3, 1, 5, 3, 1, 1, 9, 7, 5, 3, 1, 1, 5, 3, 1, 5, 3, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 9, 7, 5, 3, 1, 1, 3, 1, 1, 11, 9, 7, 5

Offset: 1

Labos Elemer, Mar 23 2001

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A020482, A049653, A060267, A060268, A060264.

with(numtheory): [seq(nextprime(2*i)-2*i,i=1..256)];

d2n[n_]:=Module[{c=2n},NextPrime[c]-c]; Array[d2n,120] (* Harvey P. Dale, May 14 2011 *)
Table[NextPrime@ # - # &[2 n], {n, 120}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = nextprime(2*n+1) - 2*n; \\ Michel Marcus, Feb 19 2017

Conjecture: Limit_{n->oo} (Sum_{k=1..n} a(k)) / (Sum_{k=1..n} log(2*k)) = 1. - Alain Rocchelli, Oct 24 2023

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

A049613 a(n) = 2n - (largest prime < 2n-2).

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5, 7, 3, 5, 3, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 9, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 9, 11, 13, 15, 3, 5, 3, 5, 7, 3, 3, 5, 7, 9, 11, 3, 3, 5, 7, 3, 5, 7, 3, 5, 3, 5, 7, 3, 5, 7, 3, 3, 5

Offset: 3

David M. Elder (elddm(AT)rhodes.edu)

			a(14)=28 - (largest prime < 26) = 28 - 23 = 5.

T. D. Noe, Table of n, a(n) for n=3..10000

Cf. A049653, A049711, A049716, A049847.

Cf. A002373, A020481, A007917.

a049613 n = 2 * n - a007917 (2 * n - 2)
-- Reinhard Zumkeller, Jan 02 2015

Table[2n-NextPrime[2n-2,-1],{n,3,100}] (* Harvey P. Dale, Aug 16 2011 *)

a(n) <= A002373(n). - R. J. Mathar, Mar 19 2008

a(n) = 2*n - A007917(2*n-2). - Reinhard Zumkeller, Jan 02 2015

A049716 a(n) = 2n + 1 - prevprime(2n + 1).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 2, 4, 2, 4, 6, 2, 4, 6, 2, 2, 4, 6, 2, 4, 2, 2, 4, 6, 2, 4, 2, 4, 6, 2, 4, 6, 8, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 6, 8, 10, 12, 14, 2, 4, 2, 4, 6, 2, 2, 4, 6, 8, 10, 2, 2, 4, 6, 2, 4, 6, 2, 4, 2, 4, 6, 2, 4, 6, 2, 2, 4

Offset: 1

N. J. A. Sloane

nonn

			n:     1  2  3  4  5  6  7  8 ...
2n+1:  3  5  7  9 11 13 15 17 ...
pp:    2  3  5  7  7 11 13 13 ...
diff:  1  2  2  2  4  2  2  4 ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A049613, A049653, A049711, A049847.

seq(2*n+1-prevprime(2*n+1), n=1..100); # Robert Israel, Jul 05 2018

Table[2n+1-NextPrime[2n+1,-1],{n,100}] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = 2*n+1-precprime(2*n); \\ Michel Marcus, Jul 06 2018

a(n) = A049711(2*n+1). - R. J. Mathar, Oct 26 2015

A049847 a(n) = (2n + 1 - prevprime(2n+1))/2.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 5, 6, 7, 1, 2, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 4

Offset: 2

N. J. A. Sloane

nonn

Cf. A049613, A049653, A049711, A049716.

Table[(2n+1-NextPrime[2n+1,-1])/2,{n,2,100}] (* Harvey P. Dale, Jul 25 2015 *)

cp's OEIS Frontend

A049711 a(n) = n - prevprime(n).

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A060308 Largest prime <= 2n.

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A060264 First prime after 2n.

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A060265 Largest prime less than 2n.

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A060266 Difference between 2n and the following prime.

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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).

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A060268 Distance of 2n from the closest prime.

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A049716 a(n) = 2n + 1 - prevprime(2n + 1).

A049847 a(n) = (2n + 1 - prevprime(2n+1))/2.