A006489 -id:A006489 - cp's OEIS frontend

A161725 Primes p such that also p+30 and p-30 are primes.

37, 41, 43, 53, 59, 67, 71, 73, 83, 97, 101, 109, 127, 137, 167, 181, 193, 197, 211, 227, 241, 263, 281, 307, 337, 367, 379, 389, 409, 419, 431, 449, 461, 479, 491, 571, 577, 587, 601, 617, 631, 643, 647, 661, 739, 757, 827, 853, 857, 907, 911, 937, 941, 967

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 17 2009

nonn

			The prime p=37 is in the sequence because 37-30=7 and 37+30=67 are also primes.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A006489, A137796, A161723, A161724, A052243.

q=6*5; lst={}; Do[p=Prime[n]; If[PrimeQ[p-q] && PrimeQ[p+q], AppendTo[lst,p]], {n, 5000}]; lst

Definition edited by Emeric Deutsch, Jun 28 2009

A268914 Minimum difference between two distinct primes whose sum is 2*prime(n), n>4.

Original entry on oeis.org

12, 12, 12, 24, 12, 24, 24, 12, 24, 48, 12, 12, 24, 36, 12, 24, 12, 36, 48, 36, 60, 24, 12, 12, 60, 48, 48, 36, 60, 24, 36, 24, 12, 72, 60, 12, 24, 36, 84, 60, 60, 84, 24, 120, 60, 96, 12, 24, 60, 24, 12, 12, 24, 84, 12, 24, 108, 48, 48, 84, 72, 72, 36, 60, 72, 36, 12, 84, 60, 12, 60, 72, 60, 48, 36, 24, 60, 24, 24, 48, 36, 48, 36, 168, 36, 48

Offset: 5

Barry Cherkas, Feb 15 2016

If p>4 is prime, any two primes that add to 2p must be equidistant from p. If p is congruent to 1 Mod 3, then p+2 and p-4 are divisible by 3. Alternatively, if p is congruent to 2 Mod 3, the p-2 and p+4 are divisible by 3. Thus, the equidistant pairs (p-2,p+2) and (p-4,p+4) cannot be primes that add to 2p. On the other hand, adding or subtracting any multiple of 6 will be congruent to the same congruence class as p and may be prime. Thus, the minimal difference between distinct primes that add to p must be a multiple of 12.

Extrapolating from computational evidence for all primes up to 10^9, we conjecture: For each multiple of 12 there are infinitely many primes p such that p-6k and p+6k are prime and 12k is the minimal difference for two distinct primes whose sum is 2p.

			For n=5, 2*prime(5)=2*11=5+17 and 17-5=12.
For n=6, 2*prime(6)=2*13=7+19 and 19-7=12.
...
For n=8, 2*prime(8)=2*19=7+31 and 31-7=24.

Barry Cherkas, Table of n, a(n) for n = 5..10003
G. H. Hardy and J. E. Littlewood, Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes, Acta Math. 44, 1-70, 1923.

Cf. A078611, A071681, A139602, A006489, A137796, A161723, A128940.

N:= 1000: # to get a(5) .. a(N)
p:= 7:
for n from 5 to N do
  p:= nextprime(p);
  for k from 6 by 6 while not isprime(p+k) or not isprime(p-k) do od:
  A[n]:= 2*k
od:
seq(A[n],n=5..N); # Robert Israel, Mar 09 2016

f[n_]:=Block[{p=Prime[n],k},k=p+6;
While[!PrimeQ[k]||!PrimeQ[2p-k],k=k+6];2(k-p)];
seq=Reap[Do[Sow[f[n]],{n,5,200}]][[2]][[1]];
seq
(*For large data sets (say, N>5000), replace 200 with N and the above algorithm is comparatively efficient.*)
Table[2 SelectFirst[Range[#/2], Function[k, AllTrue[{#/2 + k, #/2 - k}, PrimeQ]]] &[2 Prime@ n], {n, 5, 120}] (* Michael De Vlieger, Mar 09 2016, Version 10 *)

a(n) = {p = prime(n); d = 2; while (! (isprime(p-d) && isprime(p+d)),  d+=2); 2*d;} \\ Michel Marcus, Mar 17 2016

a(n) = 2*A078611(n+2).

A278869 Sophie Germain primes p such that p+6 and p-6 are primes.

Original entry on oeis.org

11, 23, 53, 173, 233, 593, 653, 1103, 1223, 2693, 2903, 2963, 4793, 5303, 6263, 6323, 7823, 9473, 10253, 11783, 12653, 13463, 15803, 20753, 25673, 27743, 27773, 29873, 31253, 33623, 38183, 38453, 39233, 40283, 41603, 44273, 44543, 54443, 54773, 59393, 60083, 62213

Offset: 1

K. D. Bajpai, Nov 29 2016

nonn

Intersection of A005384 and A006489.
After a(1), all the terms are congruent to 3 mod 10.
A prime p is Sophie Germain prime if 2*p+1 is also prime.

			11 is in the list because: 2*11 + 1 = 23 (prime), hence 11 is Sophie Germain prime; also, 11 - 6 = 5 and 11 + 6 = 17 are both prime.
23 is in the list because: 2*23 + 1 = 47 (prime), hence 23 is Sophie Germain prime; also, 23 - 6 = 17 and 23 + 6 = 29 are both prime.

K. D. Bajpai, Table of n, a(n) for n = 1..9180

Cf. A000040, A005384, A006489.

Select[Prime[Range[20000]], PrimeQ[2 # + 1] && PrimeQ[# + 6] && PrimeQ[# - 6] &]

forprime(p=1,10000, if(isprime(2*p+1) && isprime(p+6) && isprime(p-6), print1(p, ", ")))

A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)p(p + 6).

Original entry on oeis.org

935, 1729, 4301, 11339, 49321, 102131, 146969, 298351, 386389, 1089019, 1221191, 3864241, 5171489, 12640949, 16965341, 18181979, 21243961, 43974269, 51881689, 178433279, 208506509, 223626691, 230324329, 270816731, 278421569, 393806449, 849244031, 932539661

Offset: 1

Matthew Goers, Oct 25 2023

nonn

			5, 11, and 17 are primes p, p+6, p+12, called a sexy prime triple. 5*11*17 = 935, so 935 is a term.
7, 13, and 19 are the second set of sexy prime triples. 7*13*19=1729, so 1729 is the second term.

Matthew Goers, Table of n, a(n) for n = 1..1000 [a(535) corrected by _Georg Fischer_, Sep 21 2024]

Cf. A006489, A111192. Subsequence of A007304.

(#*(#^2 - 36)) & /@ Select[Prime[Range[180]], PrimeQ[# - 6] && PrimeQ[# + 6] &] (* Amiram Eldar, Oct 27 2023 *)

apply(x->x*(x-6)*(x+6), select(x->(isprime(x-6) && isprime(x) && isprime(x+6)), [1..1000])) \\ Michel Marcus, Oct 27 2023

a(n) = (A006489(n) - 6)*A006489(n)*(A006489(n) + 6).

A377014 a(n) is the number of primes p such that p - 6, p + 6 and 2*n - p are also primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 4, 1, 3, 3, 0, 4, 4, 2, 2, 3, 3, 3, 6, 3, 4, 6, 0, 5, 5, 1, 6, 4, 3, 5, 6, 4, 3, 9, 3, 2, 8, 2, 4, 7, 2, 4, 3, 3, 5, 5, 6, 4, 9, 4, 4, 11, 2, 5, 10, 1, 4, 4, 4, 4, 4, 5, 2, 7, 4, 4, 9, 2, 5, 6, 0, 6, 7, 5, 3, 6, 5, 1, 10, 7, 4, 9, 2, 5, 9, 2, 6, 5, 4, 5, 4, 4

Offset: 1

Lei Zhou, Oct 12 2024

Conjecture: a(n) = 0 only when n = 1, 2, 3, 4, 5, 6, 19, 31, 331, 499.

			a(7) = 1 since only when p = 11 are p - 6, p + 6 and 2n - p all prime.
a(12) = 3 from the cases when p is 11, 13 or 17:
  when p = 11, {p - 6, p + 6, 2n - p} = {5, 17, 13} are all prime;
  when p = 13, {p - 6, p + 6, 2n - p} = {7, 13, 19, 11} are all prime;
  when p = 17, {p - 6, p + 6, 2n - p} = {11, 17, 23, 7} are all prime.
a(19) = 0 since 2n = 38 = 7 + 31 = 19 + 19 = 31 + 7, and none of p = 7, 19, 31 can make p - 6 and p + 6 both prime.

Cf. A006489, A045917.

f:= proc(n) local i;
  nops(select(p -> andmap(isprime,[p,p-6,p+6, 2*n-p]), [seq(i,i=3..2*n,2)]))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024

m = 200; ps = {}; p = 7; While[p = NextPrime[p]; If[PrimeQ[p - 6] && PrimeQ[p + 6], AppendTo[ps, p]]; p < 2*m]; a = {}; Do[ct = 0; k = 0; While[k++; ps[[k]] < n, q = n - ps[[k]]; If[PrimeQ[q], ct++]]; AppendTo[a, ct]; If[ct == 0, AppendTo[b, n]], {n, 2, m, 2}]; a

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A161725 Primes p such that also p+30 and p-30 are primes.

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A268914 Minimum difference between two distinct primes whose sum is 2*prime(n), n>4.

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A278869 Sophie Germain primes p such that p+6 and p-6 are primes.

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A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)*p*(p + 6).

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A377014 a(n) is the number of primes p such that p - 6, p + 6 and 2*n - p are also primes.

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A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)p(p + 6).