A236119 -id:A236119 - cp's OEIS frontend

A236458 Primes p with p + 2 and prime(p) + 2 both prime.

3, 5, 17, 41, 1949, 2309, 2711, 2789, 2801, 3299, 3329, 3359, 3917, 4157, 4217, 4259, 4637, 5009, 5021, 5231, 6449, 7757, 8087, 8219, 8627, 9419, 9929, 10007, 10937, 11777, 12071, 14321, 15647, 15971, 16061, 16901, 18131, 18251, 18287, 18539

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236470, the sequence should have infinitely many terms. This is stronger than the twin prime conjecture.

See A236457 and A236467 for similar sequences.

			a(1) = 3 since 3 + 2 = 5 and prime(3) + 2 = 7 are both prime, but 2 + 2 = 4 is composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A006512, A236119, A236456, A236457, A236467, A236470.

p[n_]:=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]

s=[]; forprime(p=2, 20000, if(isprime(p+2) && isprime(prime(p)+2), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

A236457 Primes p with q = p + 2 and prime(q) + 2 both prime.

Original entry on oeis.org

3, 5, 11, 41, 107, 311, 461, 599, 641, 1277, 1619, 1997, 2309, 2381, 2789, 3671, 4787, 5099, 6659, 6701, 6827, 7457, 7487, 8219, 8537, 8597, 9929, 10709, 11117, 12071, 12107, 12251, 13709, 17747, 18047, 18251, 18521, 22091, 22637, 23027

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236456, this sequence should have infinitely many terms.

See A236458 for a similar sequence.

			a(1) = 3 since 3 + 2 = 5 and prime(5) + 2 = 13 are both prime, but 2 + 2 = 4 is not.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A006512, A236119, A236456, A236458.

p[n_]:=PrimeQ[n+2]&&PrimeQ[Prime[n+2]+2]
In[2]:= n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[2600]],AllTrue[{#+2,Prime[#+2]+2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 21 2021 *)

s=[]; forprime(p=2, 24000, q=p+2; if(isprime(q) && isprime(prime(q)+2), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

Conjecture: a(n) > 0 for all n > 31.

This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair.

			a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime.
a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime.

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

Cf. A000010, A000040, A001359, A006512, A014574, A234694, A234695, A235924, A236074, A236119.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A236467 Primes p with p + 2 and prime(p) - 2 both prime.

Original entry on oeis.org

3, 11, 29, 149, 179, 191, 269, 347, 431, 461, 617, 659, 1031, 1619, 1931, 3467, 3527, 4799, 6569, 6689, 7349, 7877, 9011, 9767, 11117, 12611, 13691, 13901, 14549, 16067, 16139, 16451, 16631, 17489, 17681, 18911, 20981, 22367, 23909, 24179

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236468, this sequence should have infinitely many terms.

See A236457 and A236458 for similar sequences.

			a(1) = 3 since 3, 3 + 2 = 5 and prime(3) - 2 = 3 are all prime, but 2 + 2 = 4 is composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A006512, A236119, A236457, A236458, A236462, A236464, A236468.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[Prime[n]-2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[3000]],AllTrue[{#+2,Prime[#]-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 11 2020 *)

A236193 Primes p with prime(p)^2 + (2p)^2 and p^2 + (2prime(p))^2 both prime.

Original entry on oeis.org

3, 139, 179, 233, 491, 929, 1217, 1429, 1597, 1613, 1987, 2243, 3061, 3499, 3529, 4507, 5737, 5779, 6329, 7247, 7823, 8263, 8839, 9941, 10259, 11317, 11383, 12157, 12421, 13093, 13219, 13367, 14449, 14669, 15101, 15877, 17449, 18523, 18593, 19051

Offset: 1

Zhi-Wei Sun, Jan 20 2014

nonn

By part (i) of the conjecture in A236192, this sequence should have infinitely many terms.

			a(1) = 3 since prime(2)^2 + (2*2)^2 = 25 is composite, but prime(3)^2 + (2*3)^2 = 5^2 + 6^2 = 61 and 3^2 + (2*prime(3))^2 = 3^2 + 10^2 = 109 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A236119, A236143, A236192.

p[n_]:=PrimeQ[Prime[n]^2+(2*n)^2]&&PrimeQ[n^2+(2*Prime[n])^2]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]

A236687 Primes p such that prime(p^2) - 2 is also prime.

Original entry on oeis.org

2, 11, 17, 47, 61, 137, 163, 229, 239, 263, 317, 389, 419, 449, 467, 557, 571, 617, 619, 653, 709, 937, 953, 1009, 1033, 1087, 1123, 1129, 1181, 1249, 1481, 1831, 1987, 2003, 2099, 2207, 2381, 2441, 2579, 2663, 2707, 3109, 3457, 3833, 4013, 4463, 4519, 4783

Offset: 1

K. D. Bajpai, Jan 29 2014

nonn

			17 is prime and appears in the sequence because prime(17^2) = 1879 and 1879 - 2 = 1877, which is also prime.
47 is prime and appears in the sequence because prime(47^2) = 19471 and 19471 - 2 = 19469, which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..564

Cf. A000040, A234695, A236119.

KD := proc() local a,b,d; a:=ithprime(n); b:=ithprime(a^2)-2; if isprime (b) then RETURN (a);fi; end: seq(KD(), n=1..500);

default(primelimit,2^31)
s=[]; forprime(p=2, 5000, if(isprime(prime(p^2)-2), s=concat(s, p))); s \\ Colin Barker, Jan 30 2014

A236688 Primes p such that prime(p^2) + 2 is also prime.

Original entry on oeis.org

7, 53, 83, 107, 149, 223, 367, 509, 701, 769, 853, 971, 1039, 1229, 1283, 1327, 1373, 1381, 1439, 1447, 1459, 1783, 1873, 1973, 2237, 2243, 2269, 2339, 2347, 2437, 2459, 2521, 2531, 2797, 2857, 3001, 3391, 3413, 3461, 3583, 3593, 3631, 3659, 3769, 3889, 3947

Offset: 1

K. D. Bajpai, Jan 29 2014

nonn

			7 is prime and appears in the sequence: prime(7^2) = 227 and 227+2 = 229, which is also prime.
53 is prime and appears in the sequence: prime(53^2) = 25469 and 25469+2 = 25471, which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1119

Cf. A000040, A234695, A236119.

KD := proc() local a,b; a:=ithprime(n); b:=ithprime(a^2)+2; if isprime (b) then RETURN (a);fi; end: seq(KD(), n=1..700);

Select[Prime[Range[600]],PrimeQ[Prime[#^2]+2]&] (* Harvey P. Dale, Aug 29 2021 *)

default(primelimit,2^31)
s=[]; forprime(p=2, 4000, if(isprime(prime(p^2)+2), s=concat(s, p))); s \\ Colin Barker, Jan 30 2014

A236143 Odd primes p with prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) both prime.

Original entry on oeis.org

7, 11, 31, 67, 179, 193, 197, 281, 347, 349, 563, 599, 757, 1123, 1453, 1543, 1933, 1987, 2083, 2531, 2971, 3037, 3259, 3547, 3583, 3701, 3919, 4027, 4483, 5023, 5581, 5591, 5647, 5981, 6449, 7207, 7297, 7603, 8291, 9049

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

By part (i) of the conjecture in A236138, this sequence should have infinitely many terms.

			a(1) = 7 with prime(6) - 6 = 13 - 6 = 7 and prime(6) - 2*prime(3) = 13 - 2*5 = 3 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A234695, A235925, A236075, A236119, A236138.

PQ[n_]:=n>0&&PrimeQ[n]
p[n_]:=PrimeQ[Prime[n-1]-(n-1)]&&PQ[Prime[n-1]-2*Prime[(n-1)/2]]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,2,10^5}]

s=[]; forprime(p=3, 10000, if(isprime(prime(p-1)-(p-1)) && isprime(prime(p-1)-2*prime((p-1)/2)), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014

A237423 Primes p such that prime(prime(p^2)) - 2 is also prime.

Original entry on oeis.org

13, 17, 167, 179, 211, 223, 337, 373, 541, 661, 743, 751, 1063, 1129, 1217, 1607, 1697, 1741, 1913, 2017, 2039, 2083, 2293, 2389, 2447, 2459, 2543, 2677, 2693, 2711, 2851, 2909, 3083, 3191, 3209, 3259, 3571, 3889, 3917

Offset: 1

K. D. Bajpai, Feb 07 2014

			13 is prime and appears in the sequence because  prime(prime(13^2)) - 2  = 8009 which is also prime.
17 is prime and appears in the sequence because  prime(prime(17^2)) - 2  = 16139 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..155

Cf. A000040, A234695, A236119, A236687, A236688, A237283

KD := proc() local a,b;  a:=ithprime(n);  b:=ithprime(ithprime(a^2))-2;  if isprime (b) then RETURN (a); fi;  end: seq(KD(), n=1..500);

p[n_] := PrimeQ[Prime[Prime[n^2]] - 2]; n = 0; Do[If[p[Prime[m]], n = n + 1; Print[n, " ", Prime[m]]], {m, 1000}] (* Bajpai *)
Select[Prime[Range[105]], PrimeQ[Prime[Prime[#^2]] - 2] &] (* Wouter Meeussen, Feb 09 2014 *)

A238185 Primes p such that prime(prime(p^2)) + 2 is also prime.

Original entry on oeis.org

2, 23, 97, 163, 463, 491, 557, 659, 677, 977, 1033, 1151, 1187, 1429, 1439, 1511, 1579, 1663, 1933, 2111, 2113, 2141, 2381, 2969, 3301, 3491, 3803, 3929, 4201, 4421, 4447, 4513, 4547, 4789, 5039, 5273, 5281, 5303, 5309, 5449, 5669, 5741, 5939, 5981, 6053

Offset: 1

K. D. Bajpai, Feb 19 2014

			23 is in the sequence because 23 is prime and prime(prime(23^2)) + 2 = 35803 is also prime.
97 is in the sequence because 97 is prime and prime(prime(97^2)) + 2 = 1269643 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..163

Cf. A000040, A234695, A236119, A236687, A236688, A237283.

KD := proc() local a,b,d; a:=ithprime(n); b:=ithprime(ithprime(a^2))+2; if isprime (b) then RETURN (a);fi; end: seq(KD(), n=1..300);

n=0; Do[If[PrimeQ[Prime[Prime[Prime[k]^2]]+2],n=n+1; Print[n," ",Prime[k]]], {k,1,5000}]
Select[Prime[Range[800]],PrimeQ[Prime[Prime[#^2]]+2]&] (* Harvey P. Dale, Dec 19 2014 *)

cp's OEIS Frontend

A236458 Primes p with p + 2 and prime(p) + 2 both prime.

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A236457 Primes p with q = p + 2 and prime(q) + 2 both prime.

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A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

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A236467 Primes p with p + 2 and prime(p) - 2 both prime.

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A236193 Primes p with prime(p)^2 + (2*p)^2 and p^2 + (2*prime(p))^2 both prime.

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A236687 Primes p such that prime(p^2) - 2 is also prime.

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A236688 Primes p such that prime(p^2) + 2 is also prime.

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A236143 Odd primes p with prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) both prime.

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A236193 Primes p with prime(p)^2 + (2p)^2 and p^2 + (2prime(p))^2 both prime.