|
%I #23 Apr 26 2022 13:54:30
%S 229,239,241,257,269,271,277,281,439,443,463,467,479,499,613,641,653,
%T 661,673,677,683,691,811,823,839,863,881,10111,10151,10169,10181,
%U 10243,10247,10253,10267,10303,10313,10331,10343,10391,10429,10453,10457
%N Primes which when added to their own rotation yield a prime.
%C Let rotation rot(k) of a number k be defined by swapping the blocks of the first [d/2] and of the last [d/2] digits of k, where d=A055642(k). If the number of digits in k is odd, the center digit remains untouched during rotation.
%C So for example the rotation of 1234 is 3412, while the rotation of 12345 is 45312.
%C Rotation differs from reversal (A004086) for numbers with at least 4 digits, that is, after A004087(168) if we are concerned with primes.
%C The sequence lists primes p such that p+rot(p) is (again) prime.
%C Differs from A061783, where rot(k) is replaced by reverse(k), from the 5-digit terms on. - _M. F. Hasler_, Mar 03 2011
%C a(n) has an odd number of digits (see RJM comment in A086004). If a(n) has 2m+1 digits, then the m-th digit of a(n) is even as otherwise a(n) + rot(a(n)) is even. - _Chai Wah Wu_, Aug 19 2015
%H Chai Wah Wu, <a href="/A086002/b086002.txt">Table of n, a(n) for n = 1..10000</a>
%e a(100)=12917 because (i) 12917 is prime and (ii) rotate(12917) = 17912 and 12917+17912=30829, which is also prime.
%p A055642 := proc(n) max(1,1+ilog10(n)) ; end:
%p rot := proc(n) local d,dl,dh,pre,suf ; d := A055642(n) ; dl := floor( d/2) ; dh := floor( (d+1)/2) ; pre := floor(n/10^dh) ; suf := n mod 10^dl ; if dl <> dh then suf*10^dh+pre+10^dl*( floor(n/10^dl) mod 10) ; else suf*10^dh+pre ; fi; end:
%p isA086002 := proc(p) if isprime(p) then isprime(p+rot(p)) ; else false; fi; end:
%p for n from 1 to 1500 do p := ithprime(n) ; if isA086002(p) then printf("%d,",p) ; fi; od: # _R. J. Mathar_, May 27 2009
%t rot[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];If[OddQ[len],FromDigits[ Join[Take[idn,-Floor[len/2]],{idn[[(len+1)/2]]},Take[idn,Floor[len/2]]]],FromDigits[ Join[ Take[ idn,-len/2],Take[idn,len/2]]]]]; Select[Prime[Range[1500]],PrimeQ[ #+rot[#]]&] (* _Harvey P. Dale_, Apr 26 2022 *)
%Y Cf. A086003, A086004.
%K base,nonn
%O 1,1
%A Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 07 2003
%E Edited by _R. J. Mathar_, May 27 2009
|
