Antonio Gimenez - cp's OEIS frontend

A364263 Numbers j such that k=210j, 2k, k^2+k, k^2-k all are averages of twin primes.

521457, 7235406, 107647749, 123718516, 161881498, 200141522, 215640361, 269177896, 301917955, 352021648, 393455227, 580398355, 716403116, 916268861, 1000979331, 1231997208, 1289093896, 1354550305, 1471667483, 1478187348, 1485031638, 1520586133

Offset: 1

Bryce Case, Jr. and Antonio Gimenez, Jul 16 2023

nonn

A derivative sequence stemming from editorial discussion with Hugo Pfoertner, consisting of GCD-reduced elements of A363500, all of which were determined to be multiples of 210 aside from the first term.

Jason Yuen, Table of n, a(n) for n = 1..9999 (first 48 terms from Bryce Case, Jr.)
Bryce Case, Jr., a364263.go

Cf. A363500.

Go
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// See link.
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a(n) = A363500(n+1)/210. - Jason Yuen, Jun 04 2024

A363500 Numbers k between twin primes p, q where k+p and k+q are also twin primes, and kp and kq are between twin primes.

Original entry on oeis.org

6, 109505970, 1519435260, 22606027290, 25980888360, 33995114580, 42029719620, 45284475810, 56527358160, 63402770550, 73924546080, 82625597670, 121883654550, 150444654360, 192416460810, 210205659510, 258719413680, 270709718160, 284455564050, 309050171430

Offset: 1

Bryce Case, Jr. and Antonio Gimenez, Jun 05 2023

nonn

Larger twin primes are found on either side of 6x, so my highly-unoptimized code simply keeps adding 6 and performing the requisite primality checks using golang's "ProbablyPrime" function, a combination of Miller-Rabin and Baillie-PSW, accurate up to 2^64. Based on seminal work by fellow OEIS contributor Antonio Gimenez.
To generate, k = 6x.
p = k-1, q = k+1, check the primality of k+p, k+q, then check the primality of ((k*p) +/- 1) and ((k*q) +/- 1).
If k > x+1 and x > 1, then all eight primes are not divisible by x. If k > 8, then k == 0 (mod 210). - Jason Yuen, Jun 02 2024

Jason Yuen, Table of n, a(n) for n = 1..10000 (first 49 terms from Bryce Case, Jr.)
Bryce Case, Jr., a363500.go

Subsequence of A066388.

Cf. A364263.

Go
```
// See link.
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a(n) = 210*A364263(n-1) for n > 1. - Hugo Pfoertner, Jun 03 2024

A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

Original entry on oeis.org

120, 144, 186, 204, 216, 246, 288, 300, 324, 342, 414, 426, 474, 516, 528, 534, 552, 582, 624, 636, 666, 696, 714, 780, 792, 804, 816, 834, 846, 870, 894, 900, 924, 960, 1002, 1026, 1044, 1056, 1074, 1080, 1134, 1140, 1146, 1158, 1176, 1206, 1242, 1254, 1266, 1272, 1314, 1332, 1338, 1344, 1350

Offset: 1

Antonio Gimenez, Jul 05 2015

nonn

From Brian Almond, Jun 23 2020: (Start)
For every prime gap g, there is a run of consecutive a(n) of length max{[(g+2)/6]-1,0}.
Gaps between successive a(n) correspond to clusters of primes all within +- 8 of each other.  The number of primes within a gap G = a(n+1) - a(n) ranges from (G/6 - 1) to (G/6 - 1) plus the number of twin primes within the gap.
Record gaps in a(n) are 24 at a(1)=120, 42 at a(2)=144, 72 at a(10)=342 and 84 at a(1003)=14706 (the next gaps of 84 occur at a(43136164)=369008652 and a(643519601)=5244999552).  No larger record gaps exist below 10^10 (n <= 1239026836).
(End)
Define a "small-gap k-tuple" to be an admissible k-tuple with all of its gaps in {2,4,6,8}.  Every gap G = a(n+1) - a(n) >= 18 contains a small-gap k-tuple with k >= G/6 - 1 and diameter G-14, G-12 or G-10.  For example, at n=40 the gap between 1080 and 1134 contains the 9-tuple p+{0,4,6,10,16,22,30,36,42} for p=1087. - Brian Almond, Jul 25 2020

			For n=120, 120 is a multiple of 6, and both 119 and 121 are composite.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Intersection of A008588 and A099047. - Michel Marcus, Jul 06 2015

Cf. A060461.

[n: n in [6..2000 by 6] | not IsPrime(n-1) and not IsPrime(n+1)]; // Vincenzo Librandi, Jul 08 2015

Select[6*Range[500], AllTrue[# + {1, -1}, CompositeQ] &] (* Harvey P. Dale, May 21 2017 *)

select(x->!isprime(x-1)&&!isprime(x+1), vector(10^3,j,6*j) ) \\ Joerg Arndt, Jul 06 2015

a(n) = 6 * A060461(n). - Brian Almond, Jun 22 2020

A255769 Primes p such that there are a prime number of composite numbers less than p.

Original entry on oeis.org

7, 11, 23, 31, 47, 59, 67, 83, 97, 109, 137, 149, 167, 179, 197, 211, 233, 269, 331, 347, 353, 367, 389, 419, 431, 439, 587, 617, 739, 751, 829, 859, 907, 919, 977, 991, 1009, 1031, 1039, 1063, 1117, 1171, 1187, 1237, 1319, 1327, 1427, 1447, 1471, 1499, 1553, 1567, 1723, 1901, 1913, 1933, 2207, 2221, 2269, 2293, 2333

Offset: 1

Antonio Gimenez, Jul 11 2015

nonn

			There are two composite numbers less than 7, namely, 4 and 6, and 2 is prime. Therefore 7 is a member of the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A072677.

c:= proc(n) option remember; `if`(n<4, 0,
      c(n-1)+`if`(isprime(n-1), 0, 1))
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if isprime(c(p)) then break fi
      od; p
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 23 2015

fQ[n_]:=PrimeQ[n-PrimePi[n]-1];Select[Prime[Range@400],fQ[#]&] (* Ivan N. Ianakiev, Jul 12 2015 *)

is_ok(n)=my(i,k=0); for(i=2,n-1,if(bigomega(i)>1,k++)); isprime(k)&&isprime(n);
first(m)=my(i=1,v=vector(m),k=0);while(i<=m,if(is_ok(k), v[i]=k;i++);k++);v; \\ Anders Hellström, Jul 29 2015

listp(nn)=forprime(p=2, nn, if (isprime(p - primepi(p) - 1), print1(p, ", "));); \\ Michel Marcus, Aug 27 2016

list(lim)=my(v=List(),n=1); forprime(p=2,lim, if(isprime(p - n++), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 28 2016

a(16)-a(61) from Ivan N. Ianakiev, Jul 12 2015

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User: Antonio Gimenez

A364263 Numbers j such that k=210*j, 2*k, k^2+k, k^2-k all are averages of twin primes.

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A363500 Numbers k between twin primes p, q where k+p and k+q are also twin primes, and k*p and k*q are between twin primes.

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A259826 Numbers n such that n is a multiple of 6 and both n-1 and n+1 are composite.

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A255769 Primes p such that there are a prime number of composite numbers less than p.

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A364263 Numbers j such that k=210j, 2k, k^2+k, k^2-k all are averages of twin primes.

A363500 Numbers k between twin primes p, q where k+p and k+q are also twin primes, and kp and kq are between twin primes.