A028334 -id:A028334 - 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

A204100 Number of integers between successive twin primes, divided by 3.

Original entry on oeis.org

0, 1, 1, 3, 3, 5, 3, 9, 1, 9, 3, 9, 3, 1, 9, 3, 9, 3, 9, 11, 23, 3, 9, 19, 15, 9, 5, 7, 5, 49, 3, 1, 9, 7, 45, 3, 5, 3, 9, 19, 25, 15, 3, 3, 5, 35, 7, 9, 1, 39, 3, 15, 9, 7, 21, 27, 1, 17, 5, 15, 9, 17, 1, 7, 5, 3, 31, 9, 13, 9, 13, 55, 13, 21, 9, 7, 5, 19

Offset: 1

Michel Lagneau, Jan 10 2012

nonn

			a(2) = 1 because there exists three numbers 8, 9 and 10 between (5,7) and (11,13) => a(2) = 3/3 = 1.

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

Cf. A028334, A063091, A046933, A204099.

T:=array(1..100,1..2):k:=0:for n from 1 to 1000 do:p1:=ithprime(n):p2:=ithprime(n+1):if p2-p1 = 2 then k:=k+1:T[k,1]:=p1:T[k,2]:=p2:else fi:od: for p from 2 to k do:x:= T[p+1,1]- T[p,2]: printf(`%d, `,(x-1)/3):od:

Join[{0},Rest[(#[[2]]-#[[1]]-1)/3&/@Partition[Rest[Flatten[Select[ Partition[ Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&]]],2]]] (* Harvey P. Dale, Jan 10 2016 *)

a(n) = (A063091(n+1)- A063091(n)-3)/3 = A204099(n)/3

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

Original entry on oeis.org

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

			We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

The first appearances are at A038664, seconds A356221.
Diagonal of A356222.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts divisors with gapless prime indices, complement A356225.
A356226 = gapless interval lengths of prime indices, run-lengths A287170.
Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

nn=1000;
gaps=Differences[Array[Prime,nn]];
Table[Position[gaps,2*n][[n,1]],{n,Select[Range[nn],Length[Position[gaps,2*#]]>=#&]}]

A330559 a(n) = (number of primes p <= prime(n) with Delta(p) == 2 (mod 4)) - (number of primes p <= prime(n) with Delta(p) == 0 (mod 4)), where Delta(p) = nextprime(p) - p.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 15, 16, 15, 14, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 26, 27, 26, 25, 24, 25, 26, 27, 28, 29, 28

Offset: 1

N. J. A. Sloane, Dec 30 2019

nonn

Equals A330560 - A330561.
Since Delta(prime(n)) grows roughly like log n, this probably changes sign infinitely often. When is the next time a(n) is zero, or the first time a(n) < 0 (if these values exist)?
Let s = A024675, the interprimes. For each n let E(n) = number of even terms of s that are <= n, and let O(n) = number of odd terms of s that are <= n. Then a(n+1) = E(n) - O(n). That is, as we progress through s, the number of evens stays greater than the number of odds. - Clark Kimberling, Feb 26 2024

			n=6: prime(6) = 13, primes p <= 13 with Delta(p) == 2 (mod 4) are 3,5,11; primes p <= 13 with Delta(p) == 0 (mod 4) are 7,13; so a(6) = 3-2 = 1.

N. J. A. Sloane, Table of n, a(n) for n = 1..99999
StackExchange, Asymptotic Distribution of Prime Gaps in Residue Classes.

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556, A330557, A330558, A330560, A330561.

Join[{0}, Accumulate[Mod[Differences[Prime[Range[2, 100]]], 4] - 1]] (* Paolo Xausa, Feb 05 2024 *)

A330560 a(n) = number of primes p <= prime(n) with Delta(p) == 2 (mod 4), where Delta(p) = nextprime(p) - p.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 43, 44, 45, 46, 47, 47, 48, 48, 49, 50, 50, 51, 51, 51, 51, 52, 53, 54

Offset: 1

N. J. A. Sloane, Dec 30 2019

nonn

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

Sequences related to the differences between successive primes: A001223 (Delta(p)), A028334, A080378, A104120, A330556, A330557, A330558, A330559, A330561.

[#[p:p in PrimesInInterval(1,NthPrime(n))| (NextPrime(p)-p) mod 4 eq 2]:n in [1..90]]; // Marius A. Burtea, Dec 31 2019

N:= 200: # for a(1)..a(N)
P:= [seq(ithprime(i), i=1..N+1)]:
Delta:= P[2..-1]-P[1..-2] mod 4:
R:= map(charfcn[2], Delta):
ListTools:-PartialSums(R); # Robert Israel, Dec 31 2019

Accumulate[Map[Boole[Mod[#, 4] == 2]&, Differences[Prime[Range[100]]]]] (* Paolo Xausa, Feb 05 2024 *)

A352743 a(n) = Product_{k=1..n} (p(k+1) + p(k))/(p(k+1) - p(k)), where p(k) = prime(k).

Original entry on oeis.org

1, 5, 20, 120, 540, 6480, 48600, 874800, 9185400, 79606800, 2388204000, 27066312000, 527793084000, 22167309528000, 498764464380000, 8312741073000000, 155171166696000000, 9310270001760000000, 198619093370880000000, 6852358721295360000000, 493369827933265920000000

Offset: 0

Thomas Ordowski, Apr 01 2022

nonn

Conjecture: a(n) is an integer for every natural n. - Thomas Ordowski
Checked up to n = 10^4. - Amiram Eldar, Mar 30 2022
Checked up to n = 10^6. - Michael S. Branicky, Apr 01 2022
Note that (a(n)-1)/(a(n)+1) is the relativistic sum of the velocities prime(k)/prime(k+1) from k = 1 to n, in units where the speed of light c = 1. - Thomas Ordowski, Apr 05 2022
a(0) = 1, a(n) is the largest k such that b(n+1) = b(n)*(k + a(n-1))/(k - a(n-1)) is prime, where b(1) = 2. By my conjecture, b(n) = prime(n). - Thomas Ordowski, Jul 30 2025

			a(4) = ((3+2)/(3-2))*((5+3)/(5-3))*((7+5)/(7-5))*((11+7)/(11-7)) = 540.

Amiram Eldar, Table of n, a(n) for n = 0..400
Mauro Fiorentini, Ordowski (congettura di) (in Italian).

Cf. A000040, A001223, A001043, A014574, A024675, A028334, A081411, A386311.

a:= proc(n) option remember; (p-> `if`(n=0, 1,
      a(n-1)*(p(n+1)+p(n))/(p(n+1)-p(n))))(ithprime)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 01 2022

p = Prime[Range[21]]; FoldList[Times, 1, (Rest[p] + Most[p])/(Rest[p] - Most[p])] (* Amiram Eldar, Apr 01 2022 *)

a(n) = my(v=primes(n+1)); prod(k=1, n, (v[k+1]+v[k])/(v[k+1]-v[k])); \\ Michel Marcus, Apr 10 2025

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    n, an, p, pp = 0, 1, 2, 3
    while True:
        yield an
        q, r = divmod(an*(pp+p), pp-p)
        assert r == 0, ("Counterexample", n, p, pp)
        n, an, p, pp = n+1, q, pp, nextprime(pp)
print(list(islice(agen(), 21))) # Michael S. Branicky, Apr 01 2022

a(n) = Product_{k=1..n} A001043(k)/A001223(k).
a(n+1) = 5 * Product_{k=1..n} A024675(k)/A028334(k+1).
Note that A024675(k) and A028334(k+1) are relatively prime.
For n >= 2, a(n) <= (prime(n)+1)*a(n-1). - Thomas Ordowski, Jul 30 2025

More terms from Amiram Eldar, Apr 01 2022

A100820 Number of odd numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 2, 0, 2, 1, 0, 2, 1, 2, 3, 1, 0, 1, 0, 1, 6, 1, 2, 0, 4, 0, 2, 2, 1, 2, 2, 0, 4, 0, 1, 0, 5, 5, 1, 0, 1, 2, 0, 4, 2, 2, 2, 0, 2, 1, 0, 4, 6, 1, 0, 1, 6, 2, 4, 0, 1, 2, 3, 2, 2, 1, 2, 3, 1, 3, 4, 0, 4, 0, 2, 1, 2, 3, 1, 0, 1, 5, 3, 1, 3, 1, 2, 5, 0, 8, 2, 4, 2, 2, 0, 2

Offset: 1

Giovanni Teofilatto, Jan 06 2005

			a(2)=0 because between 3 and 5 there are no odd numbers.
a(3)=0 because between 5 and 7 there are no odd numbers.

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Cf. A000040, A001223, A028334, A036263.

[0] cat [(NthPrime(n+1)-NthPrime(n))/2-1 : n in [2..100]]; // Wesley Ivan Hurt, Jun 01 2016

P:= select(isprime,[seq(i,i=3..1000,2)]):
0,op(map(`-`,1/2*(P[2..-1]-P[1..-2]),1)); # Robert Israel, Jun 01 2016

Table[Floor[Max[(Prime[n + 1] - Prime[n])/2 - 1, 0] ], {n, 120}] (* Ray Chandler, Jan 09 2005 *)

a(n) = (prime(n+1)-prime(n))/2-1 = A001223(n)/2-1 for n>=2. - Robert Israel, Jun 01 2016

a(n) = A028334(n) - 1 for n>=2. - Michel Marcus, Jan 04 2023

Corrected and extended by Ray Chandler, Jan 09 2005

A225423 Primes p such that p + 70000000 is also prime.

Original entry on oeis.org

37, 43, 103, 157, 181, 283, 331, 379, 409, 433, 613, 631, 643, 691, 739, 811, 823, 829, 991, 1021, 1093, 1171, 1201, 1237, 1249, 1279, 1297, 1381, 1483, 1741, 1759, 1777, 1873, 1879, 2011, 2017, 2131, 2221, 2239, 2269, 2341, 2377, 2467, 2473, 2551, 2659, 2791

Offset: 1

T. D. Noe, May 15 2013

nonn

Yitang (Tom) Zhang announced yesterday that he could prove that there are an infinite number of k-twin primes p, p+k, for some k < 70000000. Here 70000000 is only an upper bound. See the Comments in A028334. - corrected by Jonathan Sondow, May 17 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Christian Perfect and Peter Rowlett, Primes gotta stick together
Wikipedia, Yitang Zhang

Cf. A001359 (first member of a twin prime pair), A028334.

Select[Prime[Range[600]], PrimeQ[70000000 + #] &]

A279765 Primes p such that p+24 and p+48 are also primes.

Original entry on oeis.org

5, 13, 19, 23, 59, 79, 83, 89, 103, 149, 233, 269, 283, 349, 373, 409, 419, 439, 443, 499, 523, 569, 593, 653, 709, 773, 829, 839, 859, 863, 929, 1039, 1069, 1259, 1279, 1399, 1423, 1559, 1699, 1753, 1823, 1949, 1979, 2039, 2063, 2089, 2113, 2309, 2333, 2393

Offset: 1

Gerhard Kirchner, Dec 18 2016

nonn

Subsequence of A033560. The triples have the form (p,p+d,p+2d). The current sequence (d=24) continues A023241 (d=6), A185022 (d=12) and A156109 (d=18). The frequencies of such triples and the triple (p, p+3±1, p+6) in A007529 do not differ very much (see table in the link "comparison of triples"). For creating the b-file I used a file of prime differences, divided by 2 (extension of A028334). For filling the table I analyzed primes up to 10^9.

Annotation: The algorithm using a file of primes or prime differences is not difficult but not as easy as using a function like isprime(n). On the other hand, such a function needs computing time which is not negligible for large numbers.

			First term: 5, 5 + 24 = 29 and 5 + 48 = 53 are all primes.

Gerhard Kirchner, Table of n, a(n) for n = 1..10000
Gerhard Kirchner, Comparison of triples

Cf. A023241, A033560, A156109, A185022.

Select[Prime@Range@500, PrimeQ[# + 24] && PrimeQ[# + 48] &] (* Robert G. Wilson v, Dec 18 2016 *)

is(n) = for(k=0, 2, if(!ispseudoprime(n+24*k), return(0))); 1 \\ Felix Fröhlich, Dec 26 2016

A341765 Consider gaps between successive odd primes from 3 up to prime(n+2). Let k1 be number of gaps congruent to 2 (mod 6) and let k2 be number of gaps congruent to 4 (mod 6). Then a(n) = k1 - k2.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2

Offset: 1

Artur Jasinski, Feb 19 2021

nonn

Theorem A: for all n, a(n) belongs to the set: {1,2}, for proof see A342156.

The indices n for which numbers of 1's and 2's in this sequence are equal are 2, 4, 6, 10, 12, 20, 36, 46, 48 and no other up to n=10^6.

			a(1)=1 because prime(2+1)-prime(2)=5-3=2 then the gap 2 is congruent to 2 mod 6, then k1=1 and k2=0 so k1 - k2 = 1.

Cf. A000230, A001223, A028334, A321856, A341952, A342156, A039701.

k1 = 0; k2 = 0; cc = {}; Do[
gap = Prime[n + 1] - Prime[n];
If[Mod[gap/2, 3] == 1, k1 = k1 + 1,
  If[Mod[gap/2, 3] == 2, k2 = k2 + 1]]; AppendTo[cc, k1 - k2];
If[k1 - k2 == 1, , If[k1 - k2 == 2, , Print[{n, k1 - k2}]]], {n, 2,
  105}]; cc

a(n) = {my(vp = vector(n+1, k, prime(k+1)), dp = vector(#vp-1, k, (vp[k+1] - vp[k])/2)); my(s=0); for (k=1, #dp, if ((dp[k]%3)==1, s++); if ((dp[k]%3) == 2, s--)); s;} \\ Michel Marcus, Feb 27 2021

a(n) = 3 - A039701(n+2). - Andrey Zabolotskiy, Nov 04 2024

Name edited by Andrey Zabolotskiy, Nov 04 2024

cp's OEIS Frontend

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

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A204100 Number of integers between successive twin primes, divided by 3.

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A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

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A330559 a(n) = (number of primes p <= prime(n) with Delta(p) == 2 (mod 4)) - (number of primes p <= prime(n) with Delta(p) == 0 (mod 4)), where Delta(p) = nextprime(p) - p.

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A330560 a(n) = number of primes p <= prime(n) with Delta(p) == 2 (mod 4), where Delta(p) = nextprime(p) - p.

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A352743 a(n) = Product_{k=1..n} (p(k+1) + p(k))/(p(k+1) - p(k)), where p(k) = prime(k).

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A100820 Number of odd numbers between prime(n) and prime(n+1).

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