A060265 -id:A060265 - cp's OEIS frontend

A151799 Version 2 of the "previous prime" function: largest prime < n.

2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 71, 71, 73, 73, 73, 73

Offset: 3

N. J. A. Sloane, Jun 29 2009

Version 1 of the "previous prime" function is "largest prime <= n". This produces A007917, the same sequence of numerical values, except the offset (or indexing) starts at 2 instead of 3.
Maple's "prevprime" function uses version 2.
See A007917 for references and further information.

Daniel Forgues, Table of n, a(n) for n = 3..100000

Cf. A000040, A007917, A007918, A060265, A151800.

a151799 = a007917 . (subtract 1)  -- Reinhard Zumkeller, Jul 26 2012

A151799:=n->prevprime(n): seq(A151799(n), n=3..100);

Table[NextPrime[n, -1], {n, 3, 77}] (* Jean-François Alcover, May 27 2011 *)

makelist(prev_prime(n), n, 3, 79); /* Bruno Berselli, May 20 2011 */

a(n)=precprime(n-1) \\ Charles R Greathouse IV, Jul 12 2016

from sympy import prevprime
def A151799(n):
    return prevprime(n) # Chai Wah Wu, Feb 28 2018

a(n) = A000040(A000720(n-1)). - Enrique Pérez Herrero, Jul 23 2011
a(n) = n + 1 - Sum_{k=1..n}( floor(k!^(n-1)/(n-1)!)-floor((k!^(n-1)-1)/(n-1)!) ). - Anthony Browne, May 17 2016
a(n) = A060265(floor(n/2)) for n >= 4. - Georg Fischer, Nov 29 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

A020900 Greatest k such that k-th prime < twice n-th prime.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 11, 12, 14, 16, 18, 21, 22, 23, 24, 27, 30, 30, 32, 34, 34, 37, 38, 40, 44, 46, 46, 47, 47, 48, 54, 55, 58, 59, 62, 62, 65, 66, 67, 68, 71, 72, 75, 76, 77, 78, 82, 86, 87, 88, 90, 91, 92, 95, 97, 99, 99, 100, 101, 102, 103, 106, 112

Offset: 1

Clark Kimberling

nonn

			4th prime is 7, twice the 4th prime is 14, the greatest prime < 14 is 13 that is the 6th prime, hence, a(4) = 6. - _Bernard Schott_, Feb 02 2020

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000720 (pi(n)), A100484 (2*prime(n)).
Cf. A020901, A059788, A060265.
Cf. A102820 (first differences).

PrimePi[NextPrime[#,-1]]&/@(2Prime[Range[70]]) (* Harvey P. Dale, Jul 05 2012 *)

a(n) = primepi(2*prime(n)); \\ Michel Marcus, Oct 25 2017; Feb 02 2020

a(n) = A000720(A100484(n)). - Michel Marcus, Feb 02 2020

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.

A255313 Triangle read by rows: row n contains the sums of adjacent pairs of terms in row n of A088643.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 22 2015

All terms are prime by definition of A088643.

See A255313 for sorted distinct terms and A255395 for number of distinct terms.

			.  n |                  T(n,k)                    |       A255316
. ---+--------------------------------------------+----------------------
.  1 |  3                                         | 3
.  2 |  5  3                                      | 3 5
.  3 |  7  5  3                                   | 3 5  7
.  4 |  7  5  7  5                                | 5 7
.  5 | 11  7  5  7  5                             | 5 7 11
.  6 | 13 11  7  5  7  5                          | 5 7 11 13
.  7 | 13 11 13 11  7  5  3                       | 3 5  7 11 13
.  8 | 17 13 11 13 11  7  5  3                    | 3 5  7 11 13 17
.  9 | 19 17 13 11 13 11  7  5  3                 | 3 5  7 11 13 17 19
. 10 | 19 17 19 17 13 11  7  5  7  5              | 5 7 11 13 17 19
. 11 | 23 19 17 19 17 13 11  7  5  7  5           | 5 7 11 13 17 19 23
. 12 | 23 19 17 19 23 19 13 11  7  5  7  5        | 5 7 11 13 17 19 23
. 13 | 23 19 23 19 17 23 19 11  7 11 13  7  3     | 3 7 11 13 17 19 23
. 14 | 29 23 19 23 19 17 23 19 11  7 11 13  7  3  | 3 7 11 13 17 19 23 29

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A088643, A255316, A255395, A060265.

a255313 n k = a255313_tabl !! (n-1) !! (k-1)
a255313_row n = a255313_tabl !! (n-1)
a255313_tabl = zipWith (zipWith (+)) tss $ map tail tss
               where tss = tail a088643_tabl

(* A is A088643 *)
A[n_, 1] := n;
A[n_, k_] := A[n, k] = For[m = n-1, m >= 1, m--, If[PrimeQ[m + A[n, k-1]] && FreeQ[Table[A[n, j], {j, 1, k-1}], m], Return[m]]];
T[n_] := T[n] = 2 MovingAverage[Table[A[n+1, k], {k, 1, n+1}], {1, 1}];
Array[T, 14] // Flatten (* Jean-François Alcover, Aug 02 2021 *)

T(n,k) = A088643(n,k-1) + A088643(n,k), 1 <= k <= n;

T(n,1) = A060265(n+1);

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

A088631 Largest number m < n such that m+n is a prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 5, 8, 9, 8, 11, 10, 9, 14, 15, 14, 13, 18, 17, 20, 21, 20, 23, 22, 21, 26, 25, 24, 29, 30, 29, 28, 33, 32, 35, 36, 35, 34, 39, 38, 41, 40, 39, 44, 43, 42, 41, 48, 47, 50, 51, 50, 53, 54, 53, 56, 55, 54, 53, 52, 51, 50, 63, 62, 65, 64, 63, 68, 69, 68, 67, 66, 65, 74, 75

Offset: 2

N. J. A. Sloane, Nov 24 2003

			Adding 1,2,3,2,5 to 2,3,4,5,6 we get the primes 3,5,7,7,11.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A088633. Second column of A088643.

Cf. A010051, A060265, A007917.

a088631 n = a060265 n - n  -- Reinhard Zumkeller, Feb 22 2015

with(numtheory); A088631 := n->prevprime(2*n)-n;

a(n) = p-n where p = largest prime <= 2n-1.

a(n) = A060265(n) - n. - Reinhard Zumkeller, Feb 22 2015

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

A120303 Largest prime factor of Catalan number A000108(n).

Original entry on oeis.org

2, 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: 2

Alexander Adamchuk, Jul 13 2006

nonn

All prime numbers (except 3) are present in this sequence in their natural order with repetition. The number of repetitions is equal to A028334(n): differences between consecutive primes, divided by 2. - Alexander Adamchuk, Jul 30 2006
For p>3 a((p+1)/2) = p and all a(n) = p for n >= (p+1)/2 until the first occurrence of the next prime q = NextPrime(p) at a((q+1)/2) = q. - Alexander Adamchuk, Dec 27 2013
For n>2, a(n) is the largest prime less than 2*n. - Gennady Eremin, Mar 02 2021

			G.f. = 2*x^2 + 5*x^3 + 7*x^4 + 7*x^5 + 11*x^6 + 13*x^7 + 13*x^8 + 17*x^9 + ...

Gennady Eremin, Table of n, a(n) for n = 2..800
Gennady Eremin, Factoring Catalan numbers, arXiv:1908.03752 [math.NT], 2019.

Cf. A000108, A006530, A020482, A028334, A060265, A060308, A152765.

Table[Max[FactorInteger[(2n)!/n!/(n+1)! ]],{n,2,100}]
FactorInteger[CatalanNumber[#]][[-1,1]]&/@Range[2,70] (* Harvey P. Dale, May 02 2017 *)

a(n) = vecmax(factor(binomial(2*n, n)/(n+1))[,1]); \\ Michel Marcus, Nov 14 2015

a(n)=if(n>2,precprime(2*n),2) \\ Charles R Greathouse IV, Nov 17 2015

from gmpy2 import is_prime
A120303 = [2]
for n in range(3, 801):
    for k in range(2*n-1, n, -2):
        if is_prime(k, n):
            A120303.append(k)
            break
for n in range(len(A120303)):
    print(n+2, A120303[n])  # Gennady Eremin, Mar 17 2021

a(n) = A060308(n) = A060265(n) for n>2.
a(n) = A006530(A000108(n)). - Michel Marcus, Nov 14 2015
G.f.: A(x) - x^2, where A(x) is the g.f. of A060265. - Gennady Eremin, Mar 02 2021

cp's OEIS Frontend

A151799 Version 2 of the "previous prime" function: largest prime < n.

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A060308 Largest prime <= 2n.

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A020900 Greatest k such that k-th prime < twice n-th 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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A255313 Triangle read by rows: row n contains the sums of adjacent pairs of terms in row n of A088643.

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A060267 Difference between 2 closest primes surrounding 2n.

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A088631 Largest number m < n such that m+n is a prime.

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