cp's OEIS Frontend

A261458 Primes which remain prime after one, two, three and four applications of the rotate-and-add operation of A086002.

%I A261458 #12 Aug 22 2015 00:50:15
%S A261458 10010905789,10028905771,10036905763,10050905749,10056905743,
%T A261458 10060905739,10070905729,10080905719,10092905707,10098905701,
%U A261458 10102905697,10106905693,10108905691,10112905687,10130905669,10160905639,10172905627,10176905623,10188905611,10190905609
%N A261458 Primes which remain prime after one, two, three and four applications of the rotate-and-add operation of A086002.
%C A261458 There are no primes that remain prime each time after 1,2,...,6 rotate-and-add operations. Proof: by the comment in A086004, such a prime p must have an odd number of digits and must remain so after 1,2,...,5 rotate-and-add operations. Let p have 2m+1 digits, and denote the first and the (m+2)-th digits as (a,b). After a rotate-and-add operation, these digits become (c,d). It is clear that c >= a+b, d >= a+b, except when a carry occur at these digits. If a carry occurred at the (m+2)-th digit, then a carry occurred at the first digit as well. In any case when a carry occurred at these digits, the number of digits is increased by 1 and thus will have even number of digits. This implies that for such a prime p, a carry did not occur after each of the 5 rotate-and-adds. The best one can do is if (a,b) = (1,0), after 4 rotate-and-adds the digits becomes (1,1), (2,2), (4,4), (8,8) or larger and thus a carry will have occurred after at most 5 rotate-and-adds, so such a prime does not exist. - _Chai Wah Wu_, Aug 21 2015
%H A261458 Chai Wah Wu, <a href="/A261458/b261458.txt">Table of n, a(n) for n = 1..542</a>
%e A261458 Applying rotate-and-add to the prime 10010905789 four times results in 15800815799, 31600631599, 63200263199, 126399526399, all of which are prime.
%Y A261458 Cf. A086002, A086003, A086004.
%K A261458 nonn,base
%O A261458 1,1
%A A261458 _Chai Wah Wu_, Aug 20 2015