The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A037074 Numbers that are the product of a pair of twin primes. 75
 15, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. Except for the first term, all entries have digital root 8. - Lekraj Beedassy, Jun 11 2004 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 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 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 Except for 15, a product of twin primes is of the form 36k^2 - 1 (cf. A136017, A002822). - Artur Jasinski, Dec 12 2007 A072965(a(n)) = 1; A072965(m) mod A037074(n) > 0 for all m. - Reinhard Zumkeller, Jan 29 2008 The number of terms less than 10^(2n) is A007508(n). - Robert G. Wilson v, Feb 08 2012 Solutions of the equation k' = 2*sqrt(k+1), where k' is the arithmetic derivative of k. - Paolo P. Lava, Oct 30 2012 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 Subset of A210503. - Paolo P. Lava, Jan 28 2013 Semiprimes of the form 4*k^2 - 1. - Vincenzo Librandi, Apr 13 2013 REFERENCES Albert A. Mullin, "Bicomposites, twin primes and arithmetic progression", Abstract 04T-11-48, Abstracts of AMS, Vol. 25, No. 4, 2004, p. 795. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A001359(n)*A006512(n). A000010(a(n))*A000203(a(n)) = (a(n)-3)*(a(n)+1). - Jonathan Vos Post, Feb 21 2006 a(n) = (A014574(n))^2 - 1. a(n+1) = (6*A002822(n))^2 - 1. - Lekraj Beedassy, Sep 02 2006 a(n) = A075369(n) - 1. - Reinhard Zumkeller, Feb 10 2015 Sum_{n>=1} 1/a(n) = A209328. - Amiram Eldar, Nov 20 2020 A000010(a(n)) == 0 (mod 8). - Darío Clavijo, Oct 26 2022 EXAMPLE a(2)=35 because 5*7=35, that is (5,7) is the 2nd pair of twin primes. MAPLE 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 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 MATHEMATICA s = Select[ Prime@ Range@170, PrimeQ[ # + 2] &]; s(s + 2) (* Robert G. Wilson v, Feb 21 2006 *) (* 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 *) Times@@@Select[Partition[Prime[Range[500]], 2, 1], Last[#]-First[#]==2&] (* Harvey P. Dale, Oct 16 2012 *) PROG (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 (Magma) [p*(p+2): p in PrimesUpTo(1000) | IsPrime(p+2)]; // Bruno Berselli, Jul 08 2011 (Magma) IsSemiprime:=func; [s: n in [1..500] | IsSemiprime(s) where s is 4*n^2-1]; // Vincenzo Librandi, Apr 13 2013 (Haskell) a037074 = subtract 1 . a075369 -- Reinhard Zumkeller, Feb 10 2015 -- Reinhard Zumkeller, Feb 10 2015, Aug 14 2011 CROSSREFS Cf. A000010, A000203, A001359, A006512, A014574, A136017, A074480 (multiplicative closure), A209328. Cf. A071700 (subsequence). Cf. A075369. Sequence in context: A074480 A194580 A210503 * A107423 A027442 A208728 Adjacent sequences: A037071 A037072 A037073 * A037075 A037076 A037077 KEYWORD nice,nonn AUTHOR Felice Russo EXTENSIONS More terms from Erich Friedman STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 5 16:06 EDT 2023. Contains 363137 sequences. (Running on oeis4.)