cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A092110 Primes p such that 2p+3 and 2p-3 are both prime.

Original entry on oeis.org

5, 7, 13, 17, 43, 53, 67, 97, 113, 127, 137, 157, 167, 193, 223, 283, 487, 547, 563, 613, 617, 643, 647, 743, 773, 937, 1033, 1187, 1193, 1277, 1427, 1453, 1483, 1543, 1583, 1627, 1663, 1733, 1847, 2027, 2143, 2297, 2393, 2437, 2467, 2477, 2503, 2617, 2843
Offset: 1

Views

Author

Zak Seidov, Feb 21 2004

Keywords

Comments

Intersection of A023204 and A063908.
All numbers in this sequence end with 3 or 7 (except the first one, which is 5). See A136191 or A136192. - Carlos Alves, Dec 20 2007

Examples

			From _K. D. Bajpai_, Sep 08 2020: (Start)
7 is a term because 2*7 + 3 = 17 and 2*7 - 3 = 11 are both prime.
13 is a term because 2*13 + 3 = 29 and 2*13 - 3 = 23 are both prime.
(End)

Links

Crossrefs

Cf. A023204, A089531, A063908, A092109.
Cf. A136191, A136192.

Programs

  • Magma
    [p: p in PrimesUpTo(10000)|IsPrime(2*p-3) and IsPrime(2*p+3)] // Vincenzo Librandi, Nov 16 2010
  • Maple
    select(p -> isprime(p) and isprime(2*p+3) and isprime(2*p-3), [seq(2*k+1, k=1..1000)]); # K. D. Bajpai, Sep 08 2020
  • Mathematica
    Select[Prime@Range@1000,PrimeQ[2#-3]&&PrimeQ[2#+3]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)

A136191 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal being 3.

Original entry on oeis.org

13, 43, 53, 113, 193, 223, 283, 563, 613, 643, 743, 773, 1033, 1193, 1453, 1483, 1543, 1583, 1663, 1733, 2143, 2393, 2503, 2843, 3163, 3413, 3433, 3793, 3823, 4133, 4463, 4483, 4523, 4603, 4673, 4813, 5443, 5743, 5953, 6073, 6133, 6163, 6553, 6733, 6863
Offset: 1

Views

Author

Carlos Alves, Dec 20 2007

Keywords

Comments

Except for p=5, the decimals in A092110 end in 3 or 7.
Theorem: If in the triple (2n-3,n,2n+3) all numbers are primes then n=5 or the decimal representation of n ends in 3 or 7. Proof: Consider Q=(2n-3)n(2n+3), by hypothesis factorized into primes. If n is prime, n=10k+r with r=1,3,7 or 9. We want to exclude r=1 and r=9. Case n=10k+1. Then Q=5(-1+6k+240k^2+800k^3) and 5 is a factor; thus 2n-3=5 or n=5 or 2n+1=5 : this means n=4 (not prime); or n=5 (included); or n=2 (impossible, because 2n-3=1). Case n=10k+9. Then Q=5(567+1926k+2160k^2+800k^3) and 5 is a factor; the arguments, for the previous case, also hold.

Crossrefs

Intersection of A092110 and A017305.
Cf. A136192.

Programs

  • Mathematica
    Select[Prime[Range[1000]],AllTrue[{2#-3,2#+3},PrimeQ]&&IntegerDigits[#][[-1]]==3&] (* James C. McMahon, Apr 30 2025 *)
  • PARI
    isok(n)  = (n % 10 == 3) && isprime(n) && isprime(2*n-3) && isprime(2*n+3); \\ Michel Marcus, Sep 02 2013

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

Original entry on oeis.org

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

Views

Author

Carlos Alves, Dec 21 2007

Keywords

Comments

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).

Links

Crossrefs

Cf. A136191, A136192, A125272.

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

Carlos Alves, Dec 21 2007

Keywords

Comments

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).

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
Showing 1-4 of 4 results.

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