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

10010905789, 10028905771, 10036905763, 10050905749, 10056905743, 10060905739, 10070905729, 10080905719, 10092905707, 10098905701, 10102905697, 10106905693, 10108905691, 10112905687, 10130905669, 10160905639, 10172905627, 10176905623, 10188905611, 10190905609

Offset

1,1

Comments

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

Links

Chai Wah Wu, Table of n, a(n) for n = 1..542

Example

Applying rotate-and-add to the prime 10010905789 four times results in 15800815799, 31600631599, 63200263199, 126399526399, all of which are prime.

Crossrefs

Cf. A086002, A086003, A086004.

Sequence in context: A244450 A293791 A208363 * A296158 A172688 A112388

Adjacent sequences: A261455 A261456 A261457 * A261459 A261460 A261461

Keyword

nonn,base

Author

Chai Wah Wu, Aug 20 2015

Status

approved