

A085331


Numbers n such that phi(rev(n))=n.


6



1, 12, 36, 192, 1992, 2016, 31067664, 39206496, 1564356432, 3937403136, 15600000432, 22871605008, 156043560432, 156439956432, 1560000000432, 1956000004392
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OFFSET

1,2


COMMENTS

rev(2*(10^k4)) = 3*(10^k3). If 10^k3 is prime, then phi(3*(10^k3)) = 2*(10^k4), so 2*(10^k4) is a term. 10^13=7 is prime, so 2*(10^14)=12 is a term, a(2). 10^23=97 is prime, so 2*(10^24)=192 is a term, a(4). 10^33=997 is prime, so 2*(10^34)=1992 is a term, a(5). 10^173 is prime, so 2*(10^174)=199999999999999992 is a term. 10^1403 is prime, so 2*(10^1404) is a term. 10^9903 is prime, so 2*(10^9904) is a term. Conjecture: sequence is infinite.  Ray Chandler, Jul 20 2003
Let f(m,n,r,t)=((9).(m).78.(0)(n).21.(9)(m))(r).(9)(t).7 where m, n, r & t are nonnegative integers; dot between numbers means concatenation and "(m)(n)" means number of m's is n. If r*t=0 & p=f(m,n,r,t) is prime then reversal(3*p) = 1.((9)(m).56.(0)(n).43.(9)(m))(r).(9)(t).2 is in the sequence. For example p1=f(0,0,0,0)=7 so reversal(3*p1) = 12 is in the sequence, p2=f(0,0,2,0)=(7821)(2).7=782178217 so reversal(3*p2) = 1.(5643)(2).2 = 1564356432 is in the sequence & p3=f(0,0,674,0) so reversal(3*p3) = 1.(5643)(674).2 is in the sequence. Primes of the form f(m,n,r,t) are a generalized form of primes of the form 10^j3 that were already related to this sequence by Ray Chandler. For all n, A085331(n) = reversal(A072395(n)).  Farideh Firoozbakht, Jan 08 2005
The list is complete through 2050000000.  Farideh Firoozbakht, Jan 15 2005
a(13) > 10^11.  Donovan Johnson, Feb 03 2012
a(17) > 10^13.  Giovanni Resta, Aug 06 2019


LINKS

Table of n, a(n) for n=1..16.


EXAMPLE

phi[{1,21,63,291,2991,6102}] = {1,12,36,192,1992,2016}


MATHEMATICA

v = {1}; Do[ If[ n == EulerPhi[ FromDigits[ Reverse[ IntegerDigits [ n ] ] ] ], v = Append[ v, n ]; Print[ v ], If[ Mod[ n, 1000000 ] == 0, Print[ n ] ] ], {n, 2, 2050000000, 2} ] (Firoozbakht)


CROSSREFS

Cf. A000010, A004086, A069215, A101700, A102278.
Sequence in context: A278583 A329859 A216381 * A225100 A058040 A130164
Adjacent sequences: A085328 A085329 A085330 * A085332 A085333 A085334


KEYWORD

nonn,base,more


AUTHOR

Labos Elemer, Jul 04 2003


EXTENSIONS

The terms 31067664, 39206496, 1564356432 are from Farideh Firoozbakht, Jan 08 2005
a(10)a(12) from Donovan Johnson, Feb 03 2012
a(13)a(16) from Giovanni Resta, Aug 06 2019


STATUS

approved



