A125272 -id:A125272 - cp's OEIS frontend

A136204 Primes p such that 3p-2 and 3p+2 are primes (see A125272) and its decimal representation ends in 7.

7, 37, 127, 167, 257, 337, 757, 797, 887, 1307, 1597, 1657, 1667, 2347, 2557, 2897, 2927, 3067, 4297, 4327, 4877, 5087, 5147, 5227, 5417, 5857, 6337, 6827, 6917, 6967, 7127, 7187, 7547, 7687, 7867, 7877, 8147, 8447, 8527, 8647, 9857, 10037, 10687

Offset: 1

Carlos Alves, Dec 21 2007

Theorem: If in the triple (3n-2,n,3n+2) all numbers are primes, then n=5 or the decimal representation of n ends in 3 or 7. Proof: Similar to A136191. Alternative Mathematica proof: Table[nn = 10k + r; Intersection (AT)(AT) (Divisors[CoefficientList[(3nn - 2) nn(3nn + 2), k]]), {r, 1, 9, 2}]; This gives {{1, 5}, {1}, {1, 5}, {1}, {1, 5}}. Therefore only r=3 and r=7 allow nontrivial divisors (excluding nn=5 itself).

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

Cf. A136191, A136192, A125272.

filter:= proc(n) isprime(n) and isprime(3*n-2) and isprime(3*n+2) end proc:
select(filter, [seq(i,i=7..10^4,10)]); # Robert Israel, Nov 20 2023

TPrimeQ = (PrimeQ[ # - 2] && PrimeQ[ #/3] && PrimeQ[ # + 2]) &; Select[Select[Range[100000], TPrimeQ]/3, Mod[ #, 10] == 7 &]

A133313 Primes p such that 3p-2 and 3p+2 are primes (see A125272) and its decimal representation finishes with 3.

Original entry on oeis.org

3, 13, 23, 43, 103, 163, 293, 313, 433, 523, 953, 1013, 1063, 1153, 1283, 1303, 1483, 1693, 1723, 1783, 1913, 2003, 2333, 3533, 3823, 3943, 4003, 4013, 4093, 4943, 5483, 6043, 6133, 6173, 6473, 6803, 7523, 7573, 7603, 7673, 7853, 7993, 8513, 9283, 9343

Offset: 1

Carlos Alves, Dec 21 2007

Theorem: If in the triple (3n-2,n,3n+2) all numbers are primes, then n=5 or the decimal representation of n finishes with 3 or 7. Proof: Similar to A136191. Alternative Mathematica proof: Table[nn = 10k + r; Intersection @@ (Divisors[CoefficientList[(3nn - 2) nn(3nn + 2), k]]), {r, 1, 9, 2}]; This gives {{1, 5}, {1}, {1, 5}, {1}, {1, 5}}. Therefore only r=3 and r=7 allow nontrivial divisors (excluding nn=5 itself).

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

Cf. A136204 (finishing with 7), A136191, A136192, A125272.

filter:= proc(n) isprime(n) and isprime(3*n-2) and isprime(3*n+2) end proc:
select(filter, [seq(i,i=3..10^4,10)]); # Robert Israel, Nov 20 2023

TPrimeQ = (PrimeQ[ # - 2] && PrimeQ[ #/3] && PrimeQ[ # + 2]) &; Select[Select[Range[100000], TPrimeQ]/3, Mod[ #, 10] == 3 &]
Select[Prime[Range[1200]],Mod[#,10]==3&&AllTrue[3#+{2,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 18 2019 *)

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

Original entry on oeis.org

5, 7, 13, 23, 37, 43, 73, 167, 233, 263, 433, 557, 587, 593, 607, 727, 857, 1153, 1597, 1627, 1753, 2143, 2663, 2713, 3433, 3607, 3863, 3947, 4027, 4363, 4423, 4673, 5147, 5477, 5623, 5807, 5903, 6277, 7237, 7333, 7577, 8287, 8647, 8837, 8887, 9043, 10067

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*7-10=11, 3*7+10=31,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-2)*(p+2)-2*p]&&PrimeQ[(p-2)*(p+2)+2*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-2)(#+2)+{2#,-2#},PrimeQ]&] (* Harvey P. Dale, Jan 01 2025 *)

A124098 Primes p such that 3p -+ 4 are primes.

Original entry on oeis.org

3, 5, 11, 19, 31, 59, 79, 89, 131, 151, 191, 199, 229, 241, 311, 389, 409, 431, 509, 521, 541, 661, 719, 739, 821, 971, 1109, 1151, 1181, 1451, 1549, 1669, 1759, 1801, 1949, 1951, 2011, 2039, 2069, 2089, 2111, 2131, 2341, 2411, 2671, 2699, 2791, 3001, 3181

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Cf. A125215, A125216, A125272.

[p: p in PrimesUpTo(70000)| IsPrime(3*p-4)and IsPrime(3*p+4)] // Vincenzo Librandi, Jan 29 2011

lst={}; Do[p=Prime[n]; If[PrimeQ[3*p-4]&&PrimeQ[3*p+4],AppendTo[lst,p]],{n,7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)
Select[Prime[Range[500]],AllTrue[3#+{4,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 02 2015 *)

a(n)= A125216(n)/3.

A125215 Semiprimes s such that s-/+2 are primes.

Original entry on oeis.org

9, 15, 21, 39, 69, 111, 129, 309, 381, 489, 501, 771, 879, 939, 1011, 1299, 1569, 2271, 2391, 2661, 2859, 3039, 3189, 3459, 3849, 3909, 3921, 4449, 4791, 4971, 5001, 5079, 5169, 5349, 5739, 6009, 6999, 7041, 7671, 8691, 8781, 9201, 10599, 11469, 11829

Offset: 1

Zak Seidov, Nov 24 2006

nonn

All terms are multiples of 3, a(n) = 3*A125272(n). - Zak Seidov, May 06 2013

			9 = 3^2 is a term since it is a semiprime, and both 9 - 2 = 7 and 9 + 2 = 11 are primes.

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

Cf. A001358, A125272.

Reap[Do[p=Prime[i];If[PrimeQ[p+4]&&Total[Last/@FactorInteger[p+2]]==2,Sow[p+2]],{i,2*10^3}]][[2,1]]

A164566 Primes p such that 7p-6 and 7p+6 are also prime numbers.

Original entry on oeis.org

5, 11, 19, 31, 41, 61, 71, 109, 151, 211, 229, 269, 379, 419, 431, 439, 479, 619, 641, 709, 739, 809, 839, 971, 1009, 1069, 1229, 1259, 1319, 1361, 1439, 1451, 1499, 1531, 1579, 1669, 1801, 1879, 1889, 2011, 2111, 2239, 2269, 2381, 2411, 2551, 2579, 2591

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Primes of the form A087681(k)/7, any index k.

			For p=5, both 7*5-6=29 and 7*5+6=41 are prime,
for p=11, both 7*11-6=71 and 7*11+6=83 are prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A106015, A125272, A127430, A046133, A087681, A136052, A023225.

[p: p in PrimesUpTo(3000) | IsPrime(7*p-6) and IsPrime(7*p+6)]; // Vincenzo Librandi, Apr 09 2013

lst={};Do[p=Prime[n];If[PrimeQ[7*p-6]&&PrimeQ[7*p+6],AppendTo[lst,p]], {n,6!}];lst
Select[Prime[Range[700]], And @@ PrimeQ/@{7 # + 6, 7 # - 6}&] (* Vincenzo Librandi, Apr 09 2013 *)

is(n)=isprime(n) && isprime(7*n-6) && isprime(7*n+6) \\ Charles R Greathouse IV, Mar 28 2017

A136052 INTERSECT A023225. [R. J. Mathar, Aug 20 2009]

Examples rephrased by R. J. Mathar, Aug 20 2009

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

Original entry on oeis.org

7, 17, 37, 113, 157, 227, 283, 293, 313, 347, 443, 587, 787, 883, 1063, 1097, 1237, 1303, 1327, 1427, 1567, 1723, 1933, 1973, 2087, 2347, 2467, 2687, 2777, 3457, 3593, 4447, 4703, 4793, 4967, 5737, 5827, 6317, 6607, 6793, 6857, 8297, 8563, 8803, 9433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

4*10-3*7=19, 4*10+3*7=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006

lst={};Do[p=Prime[n];If[PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,7!}];lst

A164568 Primes p such that 9p-10 and 9p+10 are prime numbers.

Original entry on oeis.org

3, 7, 11, 13, 29, 41, 53, 59, 67, 97, 109, 179, 223, 239, 263, 353, 389, 409, 461, 463, 557, 601, 613, 631, 673, 757, 773, 839, 857, 937, 967, 977, 1019, 1163, 1277, 1301, 1327, 1471, 1627, 1753, 1789, 1877, 1879, 2027, 2087, 2237, 2251, 2269, 2311, 2351

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

nonn

			9*3-10=17, 9*3+10=37, ...

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

Cf. A106015, A124098, A125272, A127430, A153768, A164566, A164567

[p: p in PrimesUpTo(2500) |IsPrime(9*p-10) and IsPrime(9*p+10)]; // Vincenzo Librandi, Jun 30 2016

filter:= n -> isprime(n) and isprime(9*n-10) and isprime(9*n+10):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jun 29 2016

lst={};Do[p=Prime[n];If[PrimeQ[9*p-10]&&PrimeQ[9*p+10],AppendTo[lst,p]],{n,2*6!}];lst
Select[Prime[Range[400]], PrimeQ[9 # - 10] && PrimeQ[9 # + 10] &] (* Vincenzo Librandi, Jun 30 2016 *)
Select[Prime[Range[400]],AllTrue[9#+{10,-10},PrimeQ]&] (* Harvey P. Dale, Dec 23 2023 *)

forprime(p=3,1e4,if(isprime(9*p-10)&&isprime(9*p+10),print1(p",")))

Edited by Charles R Greathouse IV, Nov 02 2009

A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 37, 43, 47, 53, 67, 73, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 233, 257, 263, 277, 283, 293, 313, 337, 347, 353, 373, 397, 433, 443, 467, 487, 523, 547, 563, 577, 593, 607, 613, 617, 643, 647, 653, 733, 743, 757, 773, 787, 797, 887, 907, 937, 947, 953, 977

Offset: 1

Altug Alkan and Thomas Ordowski, Mar 11 2017

nonn

Note that p - q must be <= 12. Also note that there can be corresponding prime pairs (q, p) more than one way, i.e., (7, 13), (13, 17), (29, 31): (13^2 - 7^2)/24 = (17^2 - 13^2)/24 = (31^2 - 29^2)/24 = 5.
There are no terms of A045468 > 11.
Union of {2}, A006489, A060212, A092110, and A125272. - Robert Israel, Mar 13 2017

			3 is a term since (11^2 - 7^2)/24 = 3 and 3, 7, 11 are prime numbers.

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

Cf. A006489, A060212, A092110, A124865, A124866, A125272, A283532.

select(r -> isprime(r) and ((isprime(3*r+2) and isprime(3*r-2))
  or (isprime(6*r+1) and isprime(6*r-1))
  or (isprime(2*r+3) and isprime(2*r-3))
or (isprime(r+6) and isprime(r-6))), [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 13 2017

ok[n_] := PrimeQ[n] && Block[{p, q, s = Reduce[p^2-q^2 == 24 n && p>3 && q>3, {p, q}, Integers]}, If[s === {}, False, Or @@ And @@@ PrimeQ[{p, q} /. List@ ToRules@s]]]; Select[Range@1000, ok] (* Giovanni Resta, Mar 11 2017 *)

isA124865(n) = if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4, d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0)
lista(nn) = forprime(p=2, nn, if(isA124865(24*p), print1(p", ")))

For n > 5, a(n) == {3,7} mod 10.

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.

Original entry on oeis.org

3, 5, 7, 11, 19, 29, 31, 59, 101, 139, 239, 271, 829, 1031, 1201, 1439, 1511, 1531, 2251, 2609, 3929, 4349, 4969, 5449, 5639, 5711, 5801, 5881, 5981, 6521, 6569, 6701, 6949, 6959, 8221, 8831, 9001, 9181, 9209, 9419, 9511, 9929, 10139, 10711, 11839, 11981

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*11-28=5, 3*11+28=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007

lst={};Do[p=Prime[n];If[PrimeQ[(p-4)*(p+4)-4*p]&&PrimeQ[(p-4)*(p+4)+4*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-4)(#+4)+{4#,-4#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2020 *)

cp's OEIS Frontend

A136204 Primes p such that 3p-2 and 3p+2 are primes (see A125272) and its decimal representation ends in 7.

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A133313 Primes p such that 3p-2 and 3p+2 are primes (see A125272) and its decimal representation finishes with 3.

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Maple

Mathematica

A155006 Primes p such that (p-2)*(p+2)-+2*p are primes.

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Mathematica

A124098 Primes p such that 3p -+ 4 are primes.

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Magma

Mathematica

Formula

A125215 Semiprimes s such that s-/+2 are primes.

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Mathematica

A164566 Primes p such that 7*p-6 and 7*p+6 are also prime numbers.

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A155007 Primes p such that (p-3)*(p+3)-+3*p are primes.

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Mathematica

A164568 Primes p such that 9*p-10 and 9*p+10 are prime numbers.

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Magma

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

A164566 Primes p such that 7p-6 and 7p+6 are also prime numbers.

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

A164568 Primes p such that 9p-10 and 9p+10 are prime numbers.

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.