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A054735 Sums of twin prime pairs.

%I A054735 #54 Oct 22 2024 09:48:25
%S A054735 8,12,24,36,60,84,120,144,204,216,276,300,360,384,396,456,480,540,564,
%T A054735 624,696,840,864,924,1044,1140,1200,1236,1284,1320,1620,1644,1656,
%U A054735 1716,1764,2040,2064,2100,2124,2184,2304,2460,2556,2580,2604,2640,2856,2904
%N A054735 Sums of twin prime pairs.
%C A054735 (p^q)+(q^p) calculated modulo pq, where (p,q) is the n-th twin prime pair. Example: (599^601)+(601^599) == 1200 mod (599*601). - _Sam Alexander_, Nov 14 2003
%C A054735 El'hakk makes the following claim (without any proof): (q^p)+(p^q) = 2*cosh(q arctanh( sqrt( 1-((2/p)^2) ) )) + 2cosh(p arctanh( sqrt( 1-((2/q)^2) ) )) mod p*q. - _Sam Alexander_, Nov 14 2003
%C A054735 Also: Numbers N such that N/2-1 and N/2+1 both are prime. - _M. F. Hasler_, Jan 03 2013
%C A054735 Excluding the first term, all remaining terms have digital root 3, 6 or 9. - _J. W. Helkenberg_, Jul 24 2013
%C A054735 Except for the first term, this sequence is a subsequence of A005101 (Abundant numbers) and of A008594 (Multiples of 12). - _Ivan N. Ianakiev_, Jul 04 2021
%H A054735 Reinhard Zumkeller, <a href="/A054735/b054735.txt">Table of n, a(n) for n = 1..10000</a>
%H A054735 Nicholas John Bizzell-Browning, <a href="https://bura.brunel.ac.uk/handle/2438/29960">LIE scales: Composing with scales of linear intervallic expansion</a>, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
%H A054735 El'hakk, <a href="http://web.archive.org/web/http://www.geocities.com/timeparadox/elhakk.htm">Page of the time traveler</a> [Archived copy on web.archive.org, as of Oct 28 2009.]
%F A054735 a(n) = 2*A014574(n) = 4*A040040(n) = A111046(n)/2.
%F A054735 a(n) = 12*A002822(n-1) for all n > 1. - _M. F. Hasler_, Dec 12 2019
%e A054735 a(3) = 24 because the twin primes 11 and 13 add to 24.
%p A054735 ZL:=[]:for p from 1 to 1451 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 A054735 A054735 := proc(n)
%p A054735 2*A001359(n)+2;
%p A054735 end proc: # _R. J. Mathar_, Jan 06 2013
%t A054735 Select[Table[Prime[n] + 1, {n, 230}], PrimeQ[ # + 1] &] *2 (* _Ray Chandler_, Oct 12 2005 *)
%t A054735 Total/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] (* _Harvey P. Dale_, Oct 23 2022 *)
%o A054735 (PARI) is_A054735(n)={!bittest(n,0)&&isprime(n\2-1)&&isprime(n\2+1)} \\ _M. F. Hasler_, Jan 03 2013
%o A054735 (PARI) pp=1;forprime(p=1,1482, if( p==pp+2, print1(p+pp,", ")); pp=p) \\ Following a suggestion by _R. J. Cano_, Jan 05 2013
%o A054735 (Haskell)
%o A054735 a054735 = (+ 2) . (* 2) . a001359  -- _Reinhard Zumkeller_, Feb 10 2015
%Y A054735 Cf. A001359, A006512, A014574, A040040, A111046.
%K A054735 easy,nonn
%O A054735 1,1
%A A054735 _Enoch Haga_, Apr 22 2000
%E A054735 Additional comments from _Ray Chandler_, Nov 16 2003
%E A054735 Broken link fixed by _M. F. Hasler_, Jan 03 2013