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Numbers that are the product of a pair of twin primes.
76

%I #118 Feb 04 2024 01:13:12

%S 15,35,143,323,899,1763,3599,5183,10403,11663,19043,22499,32399,36863,

%T 39203,51983,57599,72899,79523,97343,121103,176399,186623,213443,

%U 272483,324899,359999,381923,412163,435599,656099,675683,685583,736163

%N Numbers that are the product of a pair of twin primes.

%C Each entry is the product of p and p+2 where both p and p+2 are prime, i.e., the product of the lesser and greater of a twin prime pair.

%C Except for the first term, all entries have digital root 8. - _Lekraj Beedassy_, Jun 11 2004

%C The above statement follows from p > 3 => (p,p+2) = (6k-1,6k+1) => p*(p+2) = 36k^2 - 1 == 8 (mod 9), and A010888 === A010878 (mod 9). - _M. F. Hasler_, Jan 11 2013

%C Albert A. Mullin states that m is a product of twin primes iff phi(m)*sigma(m) = (m-3)*(m+1), where phi(m) = A000010(m) and sigma(m) = A000203(m). Of course, for a product of distinct primes p*q we know sigma(p*q) = (p+1)*(q+1) and if p, q, are twin primes, say q = p + 2, then sigma(p*q) = (p+1)*(q+1) = (p+1)*(p+3). - _Jonathan Vos Post_, Feb 21 2006

%C Also the area of twin prime rectangles. A twin prime rectangle is a rectangle whose sides are components of twin prime pairs. E.g., the twin prime pair (3,5) produces a 3 X 5 unit rectangle which has area 15 square units. - _Cino Hilliard_, Jul 28 2006

%C Except for 15, a product of twin primes is of the form 36k^2 - 1 (cf. A136017, A002822). - _Artur Jasinski_, Dec 12 2007

%C A072965(a(n)) = 1; A072965(m) mod A037074(n) > 0 for all m. - _Reinhard Zumkeller_, Jan 29 2008

%C The number of terms less than 10^(2n) is A007508(n). - _Robert G. Wilson v_, Feb 08 2012

%C If m is the product of twin primes, then sigma(m) = m + 1 + 2*sqrt(m + 1), phi(m) = m + 1 - 2*sqrt(m + 1). pmin(m) = sqrt(m + 1) - 1, pmax(m) = sqrt(m + 1) + 1. - _Wesley Ivan Hurt_, Jan 06 2013

%C Semiprimes of the form 4*k^2 - 1. - _Vincenzo Librandi_, Apr 13 2013

%D Albert A. Mullin, "Bicomposites, twin primes and arithmetic progression", Abstract 04T-11-48, Abstracts of AMS, Vol. 25, No. 4, 2004, p. 795.

%H T. D. Noe, <a href="/A037074/b037074.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001359(n)*A006512(n). A000010(a(n))*A000203(a(n)) = (a(n)-3)*(a(n)+1). - _Jonathan Vos Post_, Feb 21 2006

%F a(n) = (A014574(n))^2 - 1. a(n+1) = (6*A002822(n))^2 - 1. - _Lekraj Beedassy_, Sep 02 2006

%F a(n) = A075369(n) - 1. - _Reinhard Zumkeller_, Feb 10 2015

%F Sum_{n>=1} 1/a(n) = A209328. - _Amiram Eldar_, Nov 20 2020

%F A000010(a(n)) == 0 (mod 8). - _DarĂ­o Clavijo_, Oct 26 2022

%e a(2)=35 because 5*7=35, that is (5,7) is the 2nd pair of twin primes.

%p ZL:=[]: for p from 1 to 863 do if (isprime(p) and isprime(p+2) ) then ZL:=[op(ZL),(p*(p+2))]; fi; od; print(ZL); # _Zerinvary Lajos_, Mar 07 2007

%p for i from 1 to 150 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i)*ithprime(i+1)}); fi; od; # _Zerinvary Lajos_, Mar 19 2007

%t s = Select[ Prime@ Range@170, PrimeQ[ # + 2] &]; s(s + 2) (* _Robert G. Wilson v_, Feb 21 2006 *)

%t (* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {2}]; Select[ Range[750000], fQ] (* _Robert G. Wilson v_, Feb 08 2012 *)

%t Times@@@Select[Partition[Prime[Range[500]],2,1],Last[#]-First[#]==2&] (* _Harvey P. Dale_, Oct 16 2012 *)

%o (PARI) g(n) = for(x=1,n,if(prime(x+1)-prime(x)==2,print1(prime(x)*prime(x+1)","))) \\ _Cino Hilliard_, Jul 28 2006

%o (Magma) [p*(p+2): p in PrimesUpTo(1000) | IsPrime(p+2)]; // _Bruno Berselli_, Jul 08 2011

%o (Magma) IsSemiprime:=func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..500] | IsSemiprime(s) where s is 4*n^2-1]; // _Vincenzo Librandi_, Apr 13 2013

%o (Haskell)

%o a037074 = subtract 1 . a075369 -- _Reinhard Zumkeller_, Feb 10 2015

%o -- _Reinhard Zumkeller_, Feb 10 2015, Aug 14 2011

%Y Cf. A000010, A000203, A001359, A006512, A014574, A136017, A074480 (multiplicative closure), A209328.

%Y Cf. A071700 (subsequence).

%Y Cf. A075369.

%K nice,nonn

%O 1,1

%A _Felice Russo_

%E More terms from _Erich Friedman_