A261352 -id:A261352 - cp's OEIS frontend

A261361 Primes p such that 2*prime(p) + 1 = prime(q) for some prime q.

3, 13, 173, 463, 523, 823, 971, 991, 1291, 1543, 2113, 4003, 4019, 4649, 5923, 6037, 6101, 7649, 10103, 12539, 12841, 17203, 17569, 18013, 21193, 22093, 23321, 25111, 27197, 31231, 32009, 32117, 33349, 34687, 35423, 35449, 37747, 39619, 41729, 41759, 42017, 43237, 43331, 44741, 45841, 50021, 51437, 52489, 55921, 56891

Offset: 1

Zhi-Wei Sun, Aug 16 2015

nonn

Conjecture: The sequence contains infinitely many terms. In general, if a,b,c are positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1, and a+b+c is even and a is not equal to b, then there are infinitely many prime pairs {p,q} such that a*prime(p) - b*prime(q) = c.
See also A261362 for a stronger conjecture.
Recall that a prime p is called a Sophie Germain prime if 2*p+1 is also prime. A well-known conjecture states that there are infinitely many Sophie Germain primes.

			a(1) = 3 since 3 is a prime, and 2*prime(3)+1 = 2*5+1 = 11 = prime(5) with 5 prime.
a(3) = 173 since 173 is a prime, and 2*prime(173)+1 = 2*1031+1 = 2063 = prime(311) with 311 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A005384, A260886, A260888, A261352, A261354, A261362.

f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,5800}]

A261353 Least positive integer k such that prime(prime(k))prime(prime(kn)) = prime(p)+2 for some prime p.

Original entry on oeis.org

11, 2, 1, 606, 350, 166, 53, 1865, 7, 45, 1308, 68, 215, 61, 256, 13, 248, 90, 1, 1779, 796, 1, 4, 444, 650, 55, 157, 303, 82, 84, 25, 3, 1912, 621, 128, 205, 164, 1091, 61, 12, 337, 1, 303, 15, 23, 418, 212, 23, 2494, 1, 472, 771, 1, 36, 8, 46, 8, 18, 264, 22, 725, 85, 65, 231, 606, 3, 1, 43, 144, 164

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(prime(m))*prime(prime(n)) = prime(p)+2 for some prime p.

This implies that the sequence A261352 has infinitely many terms.

			a(1) = 11 since prime(prime(11))*prime(prime(11*1)) = prime(31)^2 = 127^2 = 16129 = prime(1877)+2 with 1877 prime.
a(4) = 606 since prime(prime(606))*prime(prime(606*4)) = prime(4457)*prime(21589) = 42643*244471 = 10424976853 = prime(473490161)+2 with 473490161 prime.

Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..382
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1,...,31
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A109611, A261282, A261352.

f[n_]:=Prime[Prime[n]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n, " ", k];Continue,{n,1,70}]

A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

Original entry on oeis.org

31, 191, 541, 809, 1153, 1301, 2221, 3037, 3847, 4049, 4159, 5441, 8243, 10177, 12277, 13681, 14783, 15619, 17903, 19463, 20897, 22697, 24517, 25163, 25847, 25849, 26633, 26647, 27329, 27407, 28051, 32653, 35059, 35747, 36341, 36527, 37369, 37811, 38609, 40949, 42737, 46679, 51061, 51607, 54443, 54679, 56113, 57637, 60887, 61493

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2+b*prime(p)+c = prime(q).

			a(1) = 31 since 31 is a prime, and prime(31)^2-2 = 127^2-2 = 16127 = prime(1877) with 1877 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A062326, A237413, A260120, A261281, A261352, A261361.

PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
f[k_]:=Prime[Prime[k]]^2-2
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,6200}]

A261395 Primes p such that (prime(p)-1)^2 = (prime(q)-1)*(prime(r)-1) for some pair of distinct primes q and r.

Original entry on oeis.org

13, 47, 137, 191, 193, 223, 227, 313, 701, 857, 907, 947, 991, 1009, 1069, 1291, 1531, 1889, 2281, 2411, 2447, 2647, 3181, 3389, 3539, 3593, 4093, 4099, 4409, 4481, 4603, 4721, 5557, 5647, 6581, 6793, 6869, 6961, 7211, 7349, 7523, 7723, 7753, 8461, 8537, 8543, 8807, 9137, 9241, 9281

Offset: 1

Zhi-Wei Sun, Aug 17 2015

nonn

Conjecture: Let d be any nonzero integer. Then there are infinitely many prime triples (p,q,r) with p,q,r distinct such that (prime(p)+d)^2 = (prime(q)+d)*(prime(r)+d). In other words, the set {prime(p)+d: p is prime} contains infinitely many nontrivial three-term geometric progressions.

			a(1) = 13 since (prime(13)-1)^2 = (41-1)^2 = 1600 = (5-1)*(401-1) = (prime(3)-1)*(prime(79)-1) with 13, 3, 79 all prime.
a(2) = 47 since (prime(47)-1)^2 = 210^2 = 44100 = 30*1470 = (prime(11)-1)*(prime(233)-1) with 47, 11, 233 all prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..600
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A000290, A260082, A261352, A261385.

Dv[n_]:=Divisors[n]
L[n_]:=Length[Dv[n]]
f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0;Do[Do[If[PQ[Part[Dv[f[k]^2],i]+1]&&PQ[Part[Dv[f[k]^2],L[f[k]^2]-i+1]+1],n=n+1;Print[n," ",Prime[k]];Goto[aa]];Continue,{i,1,(L[f[k]^2]-1)/2}];
Label[aa];Continue,{k,1,1150}]

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A261361 Primes p such that 2*prime(p) + 1 = prime(q) for some prime q.

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A261353 Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p.

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A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

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A261395 Primes p such that (prime(p)-1)^2 = (prime(q)-1)*(prime(r)-1) for some pair of distinct primes q and r.

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A261353 Least positive integer k such that prime(prime(k))prime(prime(kn)) = prime(p)+2 for some prime p.