A147547 Smallest n-digit number m such that phi(10^n+1)=phi(m), gcd(10^n+1,m)=1 and 10 does not divide m, or zero if there is no such m.

0, 0, 779, 9991, 90901, 990001, 9090901, 94139561, 681465373, 9898047311, 86925973487, 979104060601, 9080337988583, 95255589092561, 712493161316801, 9926748805307137, 90004044661864321, 989999011990088281, 9090909102763796801, 97910150575731744097, 713349371311332607153, 9789743000892702875281, 88299846937619669895601

Offset

1,3

Comments

It is easily seen that if m is in the sequence, then phi(m.m)=phi(m)^2 where dot denotes concatenation. So the sequence b(n)=a(n).a(n) is a subsequence of A147619 and it seems that the nonzero terms of this sequence is an infinite subsequence of A147619. If 10^n+1 is prime (n must be of the form 2^k), then a(n)=0 because in this case there is no n-digit number m such that phi(10^n+1)=10^n=phi(m).

Links

Max Alekseyev, Table of n, a(n) for n = 1..59

Example

phi(979104060601)=phi(10^12+1), gcd(10^12+1,979104060601)=1, 10 doesn't divide 979104060601 and 979104060601 is the smallest 12-digit number with these properties so a(12)=979104060601. Note that phi(979104060601.979104060601)=phi(979104060601)^2.

Mathematica

a[1]=a[2]=0; a[n_]:=(b=10^n+1; c=EulerPhi[b]; For[m=c+1, !(Mod[m, 10]>0&&GCD[m, b] ==1&&c==EulerPhi[m]), m++ ]; m); Do[Print[a[n]], {n, 12}]

Crossrefs

Cf. A147548, A147549, A147619.

Sequence in context: A200559 A231252 A341190 * A259924 A135198 A292063

Adjacent sequences: A147544 A147545 A147546 * A147548 A147549 A147550

Keyword

base,nonn

Author

Farideh Firoozbakht, Nov 07 2008

Extensions

a(13) and a(14) from Max Alekseyev, Mar 12 2009

a(15)-a(20) from Hiroaki Yamanouchi, Aug 27 2014

a(21)-a(59) from Max Alekseyev, Sep 07 2014, Dec 27 2015

Status

approved