OFFSET
1,1
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
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.]
FORMULA
a(n) = 12*A002822(n-1) for all n > 1. - M. F. Hasler, Dec 12 2019
EXAMPLE
a(3) = 24 because the twin primes 11 and 13 add to 24.
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 *)
PROG
(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
(Haskell)
a054735 = (+ 2) . (* 2) . a001359 -- Reinhard Zumkeller, Feb 10 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Enoch Haga, Apr 22 2000
EXTENSIONS
Additional comments from Ray Chandler, Nov 16 2003
Broken link fixed by M. F. Hasler, Jan 03 2013
STATUS
approved