A176131 -id:A176131 - cp's OEIS frontend

A176132 Lesser of twin primes p0 such that 30p0 -+ 1 and 30p1 -+ 1 are twin primes, p1 = p0 + 2.

41, 69257, 98909, 185681, 413069, 626009, 741467, 1064951, 1096829, 1107791, 1149917, 1582811, 1819271, 1823051, 2202311, 2221379, 2748059, 3177257, 3606287, 3958457, 4275809, 4786697, 5129039, 5903897, 6533981, 6541079

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 09 2010

nonn

			41 is a term since (41, 43) are twin primes, 30*41 - 1 = 1229 and 30*41 + 1=1231 are twin primes, and 30*43 - 1 = 1289 and 30*43 + 1 = 1291 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

Cf. A000040, A001359, A006512, A176131.

lst={};Do[p0=Prime[n];p1=Prime[n+1];If[p1-p0==2&&PrimeQ[p2=p0*30-1]&&PrimeQ[p3=p0*30+1]&&PrimeQ[p4=p1*30-1]&&PrimeQ[p5=p1*30+1],AppendTo[lst,p0]],{n,5*9!}];lst
ltpQ[{a_,b_}]:=b-a==2&&AllTrue[{30a+1,30a-1,30b+1,30b-1},PrimeQ]; Select[ Partition[ Prime[Range[450000]],2,1],ltpQ][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 27 2019 *)

A176133 Pythagorean primes p (primes of form 4k + 1) such that 6p -+ 1 are twin primes.

Original entry on oeis.org

5, 17, 137, 313, 373, 397, 577, 593, 653, 773, 1033, 1117, 1433, 1613, 2137, 2153, 2293, 2333, 2677, 3257, 3413, 3593, 3673, 4153, 4217, 4597, 4657, 5197, 5693, 5717, 6373, 6673, 6857, 6997, 7297, 8377, 8573, 9277, 9413, 9433, 10357, 10973, 11813

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 09 2010

nonn

			5 is a term since 5 = 4*1 + 1, and 6*5 - 1 = 29 and 6*5 + 1 = 31 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A001359, A002144, A006512, A176131, A176132.

lst={};Do[If[PrimeQ[p=4*n+1]&&PrimeQ[6*p-1]&&PrimeQ[6*p+1],AppendTo[lst,p]],{n,0,8!}];lst
Select[4*Range[3000]+1,AllTrue[{#,6#+1,6#-1},PrimeQ]&] (* Harvey P. Dale, May 29 2025 *)

A358202 Lower twin primes p such that 6p-1 and 6p+1 are twin primes and (p+1)/6 is prime.

Original entry on oeis.org

17, 137, 23537, 92957, 157217, 318677, 326657, 440177, 510617, 521537, 558497, 577937, 617717, 651017, 661097, 861437, 969257, 1093997, 1152077, 1168337, 1177157, 1260317, 1299917, 1356077, 1463177, 1514657, 1600097, 1617437, 1768757, 1773977, 1957937, 2065577, 2271497, 2335637, 2382557, 2450597

Offset: 1

J. M. Bergot and Robert Israel, Nov 03 2022

nonn

			a(2) = 137 is a term because 137 and 139 are twin primes, 6*137-1 = 821 and 6*137+1 = 823 are twin primes, and (137+1)/6 = 23 is a prime.

Robert Israel, Table of n, a(n) for n = 1..1000

Intersection of A060213 and A176131.

P:= select(isprime, {seq(i,i=5..2*10^7,2)}):
T:= P intersect map(`-`,P,2):
R:=T intersect map(t -> (t+1)/6, T):
sort(convert(select(t -> isprime((t+1)/6), R),list));

Select[Prime[Range[180000]], PrimeQ[# + 2] && PrimeQ[6*# - 1] && PrimeQ[6*# + 1] && PrimeQ[(# + 1)/6] &] (* Amiram Eldar, Nov 03 2022 *)
Select[Prime[Range[180000]],AllTrue[{#+2,6#+1,6#-1,(#+1)/6},PrimeQ]&] (* Harvey P. Dale, Jan 29 2023 *)

A386724 Twin primes p such that 6p+1, 6p-1 is a twin prime pair.

Original entry on oeis.org

3, 5, 7, 17, 103, 107, 137, 283, 313, 347, 1033, 2027, 3257, 3673, 4217, 4547, 5023, 9433, 9767, 11833, 14593, 15137, 15733, 18253, 19423, 20717, 20983, 23537, 25847, 26113, 28753, 32057, 32323, 33073, 35053, 37307, 38327, 39163, 43607, 44623, 46183, 46273, 47743, 48407

Offset: 1

Marc Morgenegg, Jul 31 2025

nonn

{3,5} and {5,7} are the only twin prime pairs occurring in this since (6p-1)*(6p+1)*(6p+11)*(6p+13) is always divisible by 5. Therefore the smallest possible gaps for p>7 is 4 (cousin primes).

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

Cf. A002822, A001359, A014574, A176131 (subsequence), A182481, A294731. Subset of A060212.

q:= p-> isprime(p) and ormap(isprime, [p-2, p+2]) and andmap(isprime, [6*p-1, 6*p+1]):
select(q, [2*i+1$i=1..25000])[];  # Alois P. Heinz, Jul 31 2025

Select[Prime[Range[5000]], Or @@ PrimeQ[# + {-2, 2}] && And @@ PrimeQ[6*# + {-1, 1}] &] (* Amiram Eldar, Jul 31 2025 *)

More terms from Pontus von Brömssen, Jul 31 2025

cp's OEIS Frontend

A176132 Lesser of twin primes p0 such that 30p0 -+ 1 and 30p1 -+ 1 are twin primes, p1 = p0 + 2.

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A176133 Pythagorean primes p (primes of form 4k + 1) such that 6p -+ 1 are twin primes.

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Mathematica

A358202 Lower twin primes p such that 6p-1 and 6p+1 are twin primes and (p+1)/6 is prime.

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A386724 Twin primes p such that 6p+1, 6p-1 is a twin prime pair.

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cp's OEIS Frontend

A176132 Lesser of twin primes p0 such that 30*p0 -+ 1 and 30*p1 -+ 1 are twin primes, p1 = p0 + 2.

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Author

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Programs

Mathematica

A176133 Pythagorean primes p (primes of form 4*k + 1) such that 6*p -+ 1 are twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A358202 Lower twin primes p such that 6*p-1 and 6*p+1 are twin primes and (p+1)/6 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A386724 Twin primes p such that 6p+1, 6p-1 is a twin prime pair.

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Author

Keywords

Comments

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Programs

Maple

Mathematica

Extensions

A176132 Lesser of twin primes p0 such that 30p0 -+ 1 and 30p1 -+ 1 are twin primes, p1 = p0 + 2.

A176133 Pythagorean primes p (primes of form 4k + 1) such that 6p -+ 1 are twin primes.

A358202 Lower twin primes p such that 6p-1 and 6p+1 are twin primes and (p+1)/6 is prime.