A092110 -id:A092110 - cp's OEIS frontend

A136192 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal of p being 7.

7, 17, 67, 97, 127, 137, 157, 167, 487, 547, 617, 647, 937, 1187, 1277, 1427, 1627, 1847, 2027, 2297, 2437, 2467, 2477, 2617, 2857, 2927, 3137, 3457, 3727, 4007, 4057, 4157, 5167, 5417, 5657, 6247, 6257, 7027, 7477, 7867, 8467, 8737, 8747, 9127, 9227

Offset: 1

Carlos Alves, Dec 20 2007

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.

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

Cf. A092110, A136191.

bpQ[n_]:=Last[IntegerDigits[n]]==7&&And@@PrimeQ[2n+{3,-3}]; Select[Prime[ Range[1200]],bpQ] (* Harvey P. Dale, Sep 25 2013 *)

Definition clarified by Harvey P. Dale, Sep 25 2013

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

Carlos Alves, Dec 20 2007

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.

Intersection of A092110 and A017305.

Cf. A136192.

Select[Prime[Range[1000]],AllTrue[{2#-3,2#+3},PrimeQ]&&IntegerDigits[#][[-1]]==3&] (* James C. McMahon, Apr 30 2025 *)

isok(n)  = (n % 10 == 3) && isprime(n) && isprime(2*n-3) && isprime(2*n+3); \\ Michel Marcus, Sep 02 2013

A092109 Primes p such that p+3 is a semiprime.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 59, 71, 79, 83, 103, 131, 139, 163, 191, 199, 211, 223, 251, 271, 311, 331, 359, 379, 383, 419, 443, 463, 479, 499, 523, 563, 619, 631, 659, 691, 743, 839, 859, 863, 883, 911, 919, 971, 1039, 1091, 1123, 1151, 1171, 1223, 1231, 1259

Offset: 1

Zak Seidov, Feb 21 2004

Primes p such that p-3 is semiprime are in A089531; p and 2p+3 both prime, A023204; p, 2p-3 and 2p+3 prime, A092110.
Primes p such that (p+3)/2 is prime. All these primes are congruent to 3 mod 4. - Artur Jasinski, Oct 11 2008
Subsequence of A131426. - Zak Seidov, Mar 29 2015
Subsequence of A091305. - David Radcliffe, May 22 2022

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

Cf. A023204, A089531, A063908, A092110, A115334, A131426.

IsSemiprime:=func< p | &+[ k[2]: k in Factorization(p)] eq 2 >; [p: p in PrimesUpTo(1300)| IsSemiprime(p+3)]; // Vincenzo Librandi, Feb 21 2014

select(p -> isprime(p) and isprime((p+3)/2), [seq(2*k+1,k=1..1000)]); # Robert Israel, Mar 29 2015

aa = {}; k = 3; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, Prime[n]]], {n, 1, 100}]; aa (* Artur Jasinski, Oct 11 2008 *)
Select[Prime[Range[300]],PrimeOmega[#+3]==2&] (* Harvey P. Dale, Feb 07 2018 *)

is(n)=n%2 && isprime((n+3)/2) && isprime(n) \\ Charles R Greathouse IV, Jul 12 2016

a(n) = 2*A063908(n)-3 = 4*A115334(n)+3. - Artur Jasinski, Oct 11 2008

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.

A172258 Primes p such that exactly one of the numbers 2p-3 and 2p+3 is prime.

Original entry on oeis.org

2, 3, 11, 19, 23, 29, 31, 37, 41, 47, 71, 73, 83, 89, 101, 107, 139, 173, 181, 191, 197, 199, 211, 227, 229, 233, 241, 251, 263, 269, 277, 307, 311, 317, 331, 337, 347, 349, 353, 373, 379, 383, 397, 409, 421, 431, 433, 439, 443, 457, 461, 463, 467, 503, 509

Offset: 1

Juri-Stepan Gerasimov, Jan 30 2010

nonn

			a(1)=2 because 2*2-3=1 (nonprime) and 2*2+3=7 (prime);
a(2)=29 because 2*29-3=55 (nonprime) and 2*29+3=61 (prime).

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

Cf. A000040, A092110, A131426.

a := proc (n): if isprime(n) = true and isprime(2*n-3) = true and isprime(2*n+3) = false then n elif isprime(n) = true and isprime(2*n-3) = false and isprime(2*n+3) = true then n else end if end proc: seq(a(n), n = 1 .. 700); # Emeric Deutsch, Feb 15 2010

Select[Prime[Range[100]],Total[Boole[PrimeQ[2#+{3,-3}]]]==1&] (* Harvey P. Dale, Mar 27 2021 *)

Definition edited by Emeric Deutsch, Feb 15 2010

Corrected and extended by Emeric Deutsch, Feb 15 2010

cp's OEIS Frontend

A136192 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal of p being 7.

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

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A092109 Primes p such that p+3 is a semiprime.

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

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A172258 Primes p such that exactly one of the numbers 2p-3 and 2p+3 is prime.

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