A031931 -id:A031931 - cp's OEIS frontend

A086139 Let p = A046133(n), that is, let p run through the list of primes such that p+12 is also prime (A046133); a(n) = number of primes in the interval p + 1 through p + 11 inclusive.

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

Offset: 1

Labos Elemer, Jul 29 2003

nonn

From Michael De Vlieger, Jul 30 2017: (Start)
a(n) = 0 for n = {24, 25, 44, 48, 53, 57, 62, 70, 82, 84, 89, 94, ...}.
a(n) = 1 for n = {9, 14, 18, 19, 20, 21, 22, 23, 28, 29, 30, 33, ...}.
a(n) = 2 for n = {4, 5, 6, 7, 8, 10, 11, 12, 13, 17, 26, 27, 31, ...}.
a(n) = 3 for n = {1, 2, 3, 15, 16, 96, 118, 183, 266, 570, 581, ...}.
(End)

			For n=1, we have p=5, the primes between 5 and 5+12=17 are 7,11,13, so a(1)=3.

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

Cf. A031930, A031931, A046133.

a:=[]; b:=[];
for n from 1 to 200 do if isprime(ithprime(n)+12) then
   a:=[op(a),ithprime(n)];
c:=0;
for i from 1 to 11 do if isprime(ithprime(n)+i) then c:=c+1; fi; od;
b:=[op(b),c];
fi;
od:
a; # A046133b; # this sequence

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True]; d = 12; Do[s = Prime[n]; If[PrimeQ[s+d], Print[cp[s+1,s+d-1]]], {n, 1, 1000}]
(* Second program: *)
With[{d = 12}, DeleteCases[#, -1] &@ Table[Function[p, If[PrimeQ[p + d],
Count[Range[p + 1, p + d - 1], _?PrimeQ], -1] ]@ Prime@ n, {n, 252}]]
PrimePi[#+11]-PrimePi[#+1]&/@Select[Prime[Range[400]],PrimeQ[#+12]&] (* Harvey P. Dale, Jul 30 2022 *)

Definition edited by N. J. A. Sloane, Aug 05 2017 following analysis by Michael De Vlieger, Jul 30 2017

A303214 Prime numbers whose average with the previous prime is not divisible by 2 or 3.

Original entry on oeis.org

3, 211, 223, 479, 521, 631, 673, 809, 1009, 1213, 1249, 1319, 1471, 1511, 1523, 1543, 1693, 1721, 1801, 1823, 1901, 2081, 2111, 2203, 2309, 2411, 2459, 2591, 2633, 2789, 2939, 3061, 3079, 3181, 3203, 3271, 3343, 3359, 3511, 3571, 3671, 3943, 4001, 4091, 4111

Offset: 1

Olivier Bélot, Apr 19 2018

nonn

Very similar to A031931.

3 and all prime(k+1) such that A001223(k) is divisible by 12. - Robert Israel, Jul 04 2018

			p = 223 => (p + previous_prime(p))/2 = (223 +211)/2 = 7*31;
p =  53 => (p + previous_prime(p))/2 =  (53 + 51)/2 = 52 (divisible by 2).

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

Cf. A001223, A031931.

count:= 1: Res:= 3:
p:= 3:
while count < 100 do
  q:= p; p:= nextprime(p);
  v:= (q+p)/2;
  if igcd(v,6)=1 then
    count:= count+1;
    Res:= Res, p;
  fi
od:
Res;# Robert Israel, Jul 04 2018

ok(n)={my(t=n+precprime(n-1)); n > 2 && isprime(n) && t%4 && t%3} \\ Andrew Howroyd, Jul 02 2018

2 NOT(|) (p+previous_prime(p))/2 AND 3 NOT(|) (p+previous_prime(p))/2

A355161 Primes p such that p - prevprime(p) is not a power of 2.

Original entry on oeis.org

29, 37, 53, 59, 67, 79, 89, 127, 137, 149, 157, 163, 173, 179, 191, 211, 223, 239, 251, 257, 263, 269, 277, 293, 307, 331, 337, 347, 359, 373, 379, 389, 419, 431, 439, 449, 479, 509, 521, 541, 547, 557, 563, 569, 577, 587, 593, 599, 607, 613

Offset: 1

Reed Michael Upson, Jun 21 2022

nonn

A130796(n) is prevprime(p).

A031925 and A031931 are subsequences, as 6 and 12 are not powers of 2.

Cf. A000079, A031925, A031931, A130796, A151800.

Select[Prime[Range[2, 120]], (d = # - NextPrime[#, -1]) != 2^IntegerExponent[d, 2] &] (* Amiram Eldar, Jun 22 2022 *)

isp2(n) = my(p); (n==1) || (isprimepower(n, &p) && (p==2)); \\ A000079
isok(p) = isprime(p) && !isp2(p-precprime(p-1)) \\ Michel Marcus, Jun 22 2022

from itertools import islice, count
from sympy import prime, prevprime
def A355161_gen(): # generator of terms
    return filter(lambda n:((~(m:=n-prevprime(n))+1)&m)-m, (prime(n) for n in count(2)))
A355161_list = list(islice(A355161_gen(),30)) # Chai Wah Wu, Aug 23 2022

a(n) = A151800(A130796(n)).

cp's OEIS Frontend

A086139 Let p = A046133(n), that is, let p run through the list of primes such that p+12 is also prime (A046133); a(n) = number of primes in the interval p + 1 through p + 11 inclusive.

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A303214 Prime numbers whose average with the previous prime is not divisible by 2 or 3.

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A355161 Primes p such that p - prevprime(p) is not a power of 2.

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