A054735 Sums of twin prime pairs.

8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924, 1044, 1140, 1200, 1236, 1284, 1320, 1620, 1644, 1656, 1716, 1764, 2040, 2064, 2100, 2124, 2184, 2304, 2460, 2556, 2580, 2604, 2640, 2856, 2904

Offset: 1

Author

Enoch Haga, Apr 22 2000

Keywords

• easy
• nonn

Comments

(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

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

Also: Numbers N such that N/2-1 and N/2+1 both are prime. - M. F. Hasler, Jan 03 2013

Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013

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

Examples

			a(3) = 24 because the twin primes 11 and 13 add to 24.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
• El'hakk, Page of the time traveler [Archived copy on web.archive.org, as of Oct 28 2009.]

Crossrefs

Cf. A001359, A006512, A014574, A040040, A111046.

Programs

Haskell

a054735 = (+ 2) . (* 2) . a001359  -- Reinhard Zumkeller, Feb 10 2015

Maple

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
A054735 := proc(n)
2*A001359(n)+2;
end proc: # R. J. Mathar, Jan 06 2013

Mathematica

Select[Table[Prime[n] + 1, {n, 230}], PrimeQ[ # + 1] &] *2 (* Ray Chandler, Oct 12 2005 *)
Total/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Oct 23 2022 *)

PARI

is_A054735(n)={!bittest(n,0)&&isprime(n\2-1)&&isprime(n\2+1)} \\ M. F. Hasler, Jan 03 2013

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

Formula

a(n) = 2*A014574(n) = 4*A040040(n) = A111046(n)/2.

a(n) = 12*A002822(n-1) for all n > 1. - M. F. Hasler, Dec 12 2019

Extensions

Additional comments from Ray Chandler, Nov 16 2003

Broken link fixed by M. F. Hasler, Jan 03 2013