A020482 -id:A020482 - cp's OEIS frontend

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

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

A325142 a(n) = k if (n - k, n + k) is the centered Goldbach partition of 2n if it exists and -1 otherwise.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0

Offset: 0

Peter Luschny, May 02 2019

sign

Let N = 2*n = p + q where p and q are primes. We call such a pair (p, q) a Goldbach partition of N. A centered Goldbach partition is the Goldbach partition of the form (n - k, n + k) where k >= 0 is minimal. If N has a centered Goldbach partition then a(n) is this k and otherwise -1.

According to Goldbach's conjecture, any even N = 2n > 2 has a Goldbach partition, which is necessarily of the form p = n - k, q = n + k: namely, with n = (p+q)/2 and k = (q-p)/2. - M. F. Hasler, May 02 2019

			a(162571) = 78 because 325142 = 162493 + 162649 and there is no k, 0 <= k < 78, such that (162571 - k, 162571 + k) is a Goldbach partition of 325142.

Peter Luschny, Table of n, a(n) for n = 0..10000

Cf. A045917, A047160, A066285.

Cf. A002374, A112823, A002374, A234345, A020482.

a := proc(n) local k; for k from 0 to n do
if isprime(n + k) and isprime(n - k) then return k fi od: -1 end:
seq(a(n), n=0..83);

a[n_] := Module[{k}, For[k = 0, k <= n, k++, If[PrimeQ[n+k] && PrimeQ[n-k], Return[k]]]; -1]; Table[a[n], {n, 0, 83}] (* Jean-François Alcover, Jul 06 2019, from Maple *)

a(n) = for(k=0, n, if(ispseudoprime(n+k) && ispseudoprime(n-k), return(k))); -1 \\ Felix Fröhlich, May 02 2019

apply( A325142(n)=-!forprime(p=n,2*n, isprime(n*2-p)&&return(p-n)), [0..99]) \\ M. F. Hasler, May 02 2019

a(n) = n - A112823(n) = A234345(n) - n (= n - A002374(n) for n > 2). - M. F. Hasler, May 02 2019

a(n) = A047160(n) = A066285(n)/2 for n >= 2. - Alois P. Heinz, Jun 01 2020

A104772 If n<=2 then n else (if n is odd then 2a(n+1) else pq, where n=p+q, p<=q, primes).

Original entry on oeis.org

1, 2, 8, 4, 18, 9, 30, 15, 42, 21, 70, 35, 66, 33, 78, 39, 130, 65, 102, 51, 114, 57, 190, 95, 138, 69, 230, 115, 322, 161, 174, 87, 186, 93, 310, 155, 434, 217, 222, 111, 370, 185, 246, 123, 258, 129, 430, 215, 282, 141, 470, 235, 658, 329, 318, 159, 530, 265, 742

Offset: 1

Reinhard Zumkeller, Mar 24 2005

nonn

Encoding of positive integers based on the Goldbach conjecture, see A104774 for decoding: A104774(A104772(n))=n;
a(n - n mod 2) = (2^(1 + n mod 2)) * A020481(floor(n/2))*A020482(floor(n/2));
for numbers greater than 4: a(even) = odd and a(odd) = even;
A001222(a(n)) = A010693(n) for n>2;
a(a(n)) = A104773(n).

Index entries for sequences related to Goldbach conjecture

Cf. A001358, A046315, A090967.

For k>1: a(2*k)=A020481(k)*A020482(k) and a(2*k-1)=2*a(2*k).

A377972 a(n) is the greatest i such that 2n-prime(i) is also a prime, where prime(i) is the i-th prime.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 22, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 29, 30, 30, 29, 31, 31, 32

Offset: 2

Michel Eduardo Beleza Yamagishi, Nov 13 2024

nonn

			For n=2, 2*2 - 2 = 2 and pi(2) = 1.
For n=3, 2*3 - 3 = 3, pi(3)=2.

Cf. A000040, A000720, A020482, A377758.

Table[Max[PrimePi[Flatten[Select[IntegerPartitions[2 n, {2}], AllTrue[#, PrimeQ] &]]]], {n, 2, 101}]

a(n) = forprime(q=2, n, if(isprime(2*n-q), return(primepi(2*n-q)))); \\ Michel Marcus, Nov 16 2024

a(n) = A000720(A020482(n)).

prime(a(n)) + prime(A377758(n)) = 2*n.

A159482 Greatest odd prime q < 2n such that p < q and p prime and p = 2n - q or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 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, 79, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 109, 113, 113, 109, 127, 127

Offset: 1

Pierre CAMI, Apr 14 2009

nonn

q + p = 2*n and p*q = ((q+p)/2)^2 - ((q-p)/2)^2 = n^2 - ((q-p)/2)^2.

Essentially the same as A020482. - R. J. Mathar, Oct 24 2009

Pierre CAMI, Table of n, a(n) for n = 1..55000

A210957 Prime pair (p, q), p<=q, such that p + q = 2n and pq is the minimal product.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Jun 29 2012

nonn

A020481 and A020482 interleaved.

			-----------------------------------
                 2*n    A073046(n)
       Pair       =         =
n     (p, q)     p+q       p*q
-----------------------------------
2     (2, 2)      4          4
3     (3, 3)      6          9
4     (3, 5)      8         15
5     (3, 7)     10         21
6     (5, 7)     12         35
7     (3, 11)    14         33
8     (3, 13)    16         39
9     (5, 13)    18         65
10    (3, 17)    20         51
11    (3, 19)    22         57
12    (5, 19)    24         95

Cf. A000040, A002373, A073046, A210967.

p_n = A020481(n), n >= 2.
q_n = A020482(n), n >= 2.
p_n + q_n = 2*n, n >= 2.
p_n * q_n = A073046(n), n >= 2.

A234649 Difference between the first members of the widest and the narrowest prime pair having an arithmetic mean of n.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 6, 6, 10, 8, 12, 0, 14, 14, 10, 14, 14, 16, 18, 16, 16, 12, 22, 16, 20, 24, 24, 26, 26, 28, 26, 32, 30, 26, 36, 16, 36, 36, 28, 36, 36, 18, 44, 38, 40, 44, 42, 40, 50, 48, 40, 42, 52, 30, 42, 46, 42, 56, 56, 58, 48, 60, 64, 56, 66, 60, 48, 60, 70, 68, 68, 54, 68, 74, 60, 56

Offset: 8

Ralf Stephan, Dec 29 2013

nonn

The widest prime pair with a mean of n is (A002373(n),A020482(n)) and the narrowest is (A078587(n),A078496(n)).
Existence of a(n) for all n depends on A061357(n) > 0.
Even numbers missing in the subsequence with n<10^5 are 34,62,82,88,112,116,118,122,130,140,152...
a(n) = 0 for n=4,5,6,7,19 because A061357(n) = 1.

			The prime pairs with an arithmetic mean of 18 are (17,19), (13,23), (7,29), and (5,31), so a(18) = 17-5 = 31-19 = 12. The only pair with mean of 19 is (7,31) so a(19) = 0.

Ralf Stephan, Table of n, a(n) for n = 8..100000

Cf. A045917.

a(n)=mi=0;ma=0;forprime(p=3,n-1,if(isprime(2*n-p),if(!mi,mi=2*n-p);ma=2*n-p));if(!ma,-1,mi-ma)

a(n) = A078587(n) - A002373(n) = A078496(n) - A020482(n).

A255274 From Goldbach conjecture: Consider the pairs (2n-+1, 3), (2n-1, 5), (2n-3, 7), ..., (3, 2n+1) of odd numbers having sum 2n+4; a(n) is the index of the first pair of primes (p, q) on the list.

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, 9, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 6, 5, 6, 9, 1, 2, 1, 2, 3, 1, 1, 2, 3, 5, 5, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2

Offset: 1

Michel Lagneau, Feb 20 2015

nonn

a(n) = A049847(n) for n = 1..46. The values of n such that a(n) is different from A049847(n) are 47, 59, 62, 72, 93, 102, 108, 123, 144, 149, 152, 161, 164, 171, 182, 197, 203, 207, 213, 227, ...

The corresponding pairs of primes are (3, 3), (3, 5), (3, 7), (5, 7), (3, 11), (3, 13), (5, 13), (3, 17), ... (A210957).

			a(13)=3 because 2*13 + 4 = 30 => 13 pairs (27,3), (25,5), (23,7), ..., (3,27) and the pair (23,7) is the third pair having prime elements.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A255274, Sequence Machine.

Cf. A000040, A002375, A049847, A210957, A020482.

nn:=100:for n from 6 by 2 to nn do:ii:=0:it:=1:for p from 3 by 2 to n while(ii=0) do:if type(n-p,prime)=true and type(p,prime)=true then ii:=1: printf(`%d, `,it):else it:=it+1:fi:od:od:

a(n)=my(m=2*n+4); forprime(q=3, n+2, if(isprime(m-q), return(q\2))) \\ Charles R Greathouse IV, Jan 07 2022

a(n) = n + (3-A020482(n+2))/2 = (A020481(n+2)-1)/2 via the Maiga link. - Bill McEachen, Jan 02 2022

Edited by N. J. A. Sloane, Sep 12 2017

A317595 a(n) is the number of primes between 2n and the largest prime p such that 2n-p is also a prime.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0

Offset: 2

Lei Zhou, Aug 01 2018

If the Goldbach Conjecture is true, this sequence is defined for n >= 2.

			For n=2, 2n=4 = 2+2, there is one prime, which is 3, between 2 and 4. So a(2)=1;
...
For n=8, 2n=16 = 13+3, there is no prime between 13 and 16. So a(8)=0;
...
For n=49, 2n=98 = 79+19, there are three primes, 83, 89, and 97 between 79 and 98 such that the difference of 98 and these primes, 15, 9, and 1 respectively, are not prime. So a(49)=3.

Index entries for sequences related to Goldbach conjecture

Cf. A000040, A000720, A002375, A020482.

Table[n2 = n*2; p = NextPrime[n2]; ct = 0; While[p = NextPrime[p, -1]; ! PrimeQ[n2 - p], ct++]; ct, {n, 2, 88}]

a(n) = A000720(A020482(n)) - A020482(2*n). - Michel Marcus, Aug 02 2018

A375358 a(n) is the greatest difference between m and k, with m, k both prime such that k + m = p + q, where (p, q) is the n-th twin prime pair.

Original entry on oeis.org

2, 2, 14, 26, 46, 74, 106, 134, 194, 206, 266, 286, 346, 374, 382, 442, 454, 506, 550, 614, 686, 818, 854, 914, 1034, 1118, 1186, 1226, 1274, 1294, 1606, 1630, 1618, 1702, 1754, 2018, 2042, 2078, 2102, 2174, 2290, 2434, 2546, 2534, 2582, 2626, 2846, 2890, 2950

Offset: 1

Gonzalo Martínez, Aug 13 2024

nonn

If p and q are twin primes and x is their average, then among all pairs of primes (k, m) such that |x - k| = |x - m|, it is observed that p and q are at the smallest distance from x, which is 1. Our interest lies in finding the pair (m, k) such that the distance to x is maximum and then determining |k - m|.

			Since the 3rd pair of twin primes is (11, 13), whose sum is 24, and the other pairs of primes that sum to 24 are (5, 19) and (7, 17), the greatest difference is 19 - 5 = 14. Therefore, a(3) = 14.

Cf. A001359, A020482, A014574.

from itertools import islice
from sympy import isprime, nextprime, primerange
def agen(): # generator of terms
    p, q = 2, 3
    while True:
        if q - p == 2:
            s = p + q
            yield max(m-k for k in primerange(2, s//2+1) if isprime(m:=s-k))
        p, q = q, nextprime(q)
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 13 2024

cp's OEIS Frontend

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

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A325142 a(n) = k if (n - k, n + k) is the centered Goldbach partition of 2n if it exists and -1 otherwise.

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A104772 If n<=2 then n else (if n is odd then 2*a(n+1) else p*q, where n=p+q, p<=q, primes).

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A377972 a(n) is the greatest i such that 2n-prime(i) is also a prime, where prime(i) is the i-th prime.

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A159482 Greatest odd prime q < 2*n such that p < q and p prime and p = 2*n - q or 0 if no such prime exists.

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A210957 Prime pair (p, q), p<=q, such that p + q = 2*n and p*q is the minimal product.

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A234649 Difference between the first members of the widest and the narrowest prime pair having an arithmetic mean of n.

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A255274 From Goldbach conjecture: Consider the pairs (2n-+1, 3), (2n-1, 5), (2n-3, 7), ..., (3, 2n+1) of odd numbers having sum 2n+4; a(n) is the index of the first pair of primes (p, q) on the list.

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A104772 If n<=2 then n else (if n is odd then 2a(n+1) else pq, where n=p+q, p<=q, primes).

A159482 Greatest odd prime q < 2n such that p < q and p prime and p = 2n - q or 0 if no such prime exists.

A210957 Prime pair (p, q), p<=q, such that p + q = 2n and pq is the minimal product.