A086003 -id:A086003 - cp's OEIS frontend

A086002 Primes which when added to their own rotation yield a prime.

229, 239, 241, 257, 269, 271, 277, 281, 439, 443, 463, 467, 479, 499, 613, 641, 653, 661, 673, 677, 683, 691, 811, 823, 839, 863, 881, 10111, 10151, 10169, 10181, 10243, 10247, 10253, 10267, 10303, 10313, 10331, 10343, 10391, 10429, 10453, 10457

Offset: 1

Chuck Seggelin, Jul 07 2003

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.
So for example the rotation of 1234 is 3412, while the rotation of 12345 is 45312.
Rotation differs from reversal (A004086) for numbers with at least 4 digits, that is, after A004087(168) if we are concerned with primes.
The sequence lists primes p such that p+rot(p) is (again) prime.
Differs from A061783, where rot(k) is replaced by reverse(k), from the 5-digit terms on. - M. F. Hasler, Mar 03 2011
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

			a(100)=12917 because (i) 12917 is prime and (ii) rotate(12917) = 17912 and 12917+17912=30829, which is also prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A086003, A086004.

A055642 := proc(n) max(1,1+ilog10(n)) ; end:
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:
isA086002 := proc(p) if isprime(p) then isprime(p+rot(p)) ; else false; fi; end:
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

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 *)

Edited by R. J. Mathar, May 27 2009

A086004 Primes which remain prime after one and after two and after three applications of the rotate-and-add operation of A086002.

Original entry on oeis.org

12917, 12919, 18911, 18913, 22907, 24907, 26903, 28901, 1088063, 1288043, 1408031, 1428029, 1528019, 100083679, 100280419, 100283849, 100483847, 100692793, 100880413, 101080159, 101283839, 101683093, 101683663, 102080149

Offset: 1

Chuck Seggelin, Jul 07 2003

These are the primes of A086003 which in addition remain prime after one additional, third application of the rotate-and-add operation.
Note: Have not yet found any 4-Rotation Cycle Primes.
Conjecture 1: Rotation and addition of primes with even numbers of digits never yields a prime.
Conjecture 2: There are no 5-Rotation Cycle Primes.
[Conjecture 1 is true because rotation for even numbers of the form 10^k*a+b yields 10^k*b+a, so rotation-and-add yields (10^k+1)*(a+b), which obviously contains a divisor A000533. RJM, Sep 17 2009]
4-Rotation Cycle Primes exist and are listed in A261458. - Chai Wah Wu, Aug 20 2015

			a(1)=12917 is in the sequence because 2-fold rotate-and-add yields the prime 60659 as shown in A086003, and the third application yields 60659+59660 = 120319 which still is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A086002, A086003.

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]]]]]; a3rotQ[n_]:=AllTrue[Rest[NestList[ #+rot[ #]&,n,3]],PrimeQ]; Select[Prime[Range[5880000]],a3rotQ] (* Harvey P. Dale, Apr 26 2022 *)

{p in A086003: p+rot(p) in A086003}.

Condensed by R. J. Mathar, Sep 17 2009

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

Original entry on oeis.org

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

Offset: 1

Chai Wah Wu, Aug 20 2015

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

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

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

Cf. A086002, A086003, A086004.

A327825 "Rotation" of n: swap the first [d/2] and last [d/2] digits, when n has d digits.

Original entry on oeis.org

979, 89, 189, 289, 389, 489, 589, 689, 789, 889, 989, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 10, 110, 210, 310, 410, 510, 610, 710, 810, 910, 1010, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2010, 2110, 2210, 2310, 2410, 2510, 2610, 2710, 2810, 2910, 3010, 3110, 3210

Offset: 979

M. F. Hasler, Sep 26 2019

When n has an odd number of digits, the middle one remains at its place.

This operation, denoted "rotation" k -> rot(k) in sequences A086002, A086003, A086004, is indistinguishable from A004086 (reverse n) for numbers < 1000. Therefore the offset is chosen as to have 2/3 of the displayed terms beyond this limit and 1/3 below. This makes it easy to find the sequence searching for the terms near that limit, ..., 999, 10, 110, 210,....

			a(123) = concat(3, 2, 1) = 321.
a(1234) = concat(34, 12) = 3412.
a(12345) = concat(45, 3, 12) = 45312.

Cf. A086002, A086003, A086004, A004086.

a(n)={n=digits(n); fromdigits(concat([n[#n\/2+1..#n],n[#n\2+1..#n\/2],n[1..#n\2]]))}

a(n) = A004086(n) ("reverse n") for n < 1000.

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A086002 Primes which when added to their own rotation yield a prime.

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A086004 Primes which remain prime after one and after two and after three applications of the rotate-and-add operation of A086002.

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

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A327825 "Rotation" of n: swap the first [d/2] and last [d/2] digits, when n has d digits.

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