%I #41 Jan 14 2025 13:51:18
%S 52,1230,5032,5662,10040,14450,56253,56962,58882,92944,564472,731935,
%T 1865170,10882630,102178040,127648411,18484522651,100000000040,
%U 100219780040,163129755801,1021999978040,1443229949041
%N Composite solutions to the equation reversal(x) - phi(x) = 1.
%C A prime p is a solution of reversal(x) - phi(x) = 1 iff p is a palindrome.
%C If p = 15*10^n-1 is prime then 38*p is in the sequence. The first three such terms are 38*(15*10^1-1)=a(4), 38*(15*10^2-1)=a(8), and 38*(15*10^15-1).
%C If p=25*10^m+1 is prime then 40*p is in the sequence.
%C The sequence A230020 gives composite solutions of equation sigma(x)-reversal(x)=1 and the sequence A130913 gives composite solutions of equation phi(x)+sigma(x)=2*reversal(x). - _Farideh Firoozbakht_, Nov 26 2013
%C From _Farideh Firoozbakht_, Feb 07 2014: (Start)
%C Let f(s,m,r) = 10^(r*m+4r+s+2)+22*10^((s/2)+2)*(10^(m+2)-1)*(10^(r*(m+4))-1)/(10^(m+4)-1)+40, where s, m and r are nonnegative integers.
%C If p=f(s,m,r)/40 is prime then s>1 and f(s,m,r)=40*p is in the sequence.
%C If r=0 then f(s,m,0) = 10^(s+2)+40 = 1.0(s).40, where dot "." means concatenation and x(y) means the digit x is repeated y times.
%C If r>0 then f(s,m,r) is an integer iff s is even. In that case, f(s,m,r) = 1.0(s/2).(21.9(m).78)(r).0(s/2).40.
%C Examples include a(5)=f(2,m,0)=10040, a(15)=f(2,0,1)=102178040, a(18)=f(9,m,0)=100000000040 and a(19)=f(4,1,1)=100219780040.
%C f(1970,19,19) is a 2410-digit term of the sequence. (End)
%C a(23) > 10^13. - _Giovanni Resta_, Aug 12 2019
%e reversal(52)-phi(52)=25-24=1.
%t r[n_]:=FromDigits[Reverse[IntegerDigits[n]]]; Do[c=r[n];If[c<n && c-EulerPhi[n]==1,Print[n]],{n,2100000000}]
%Y Cf. A000010, A004086, A130001, A130913, A230020, A137598.
%K nonn,base,more
%O 1,1
%A _Farideh Firoozbakht_, Apr 28 2007, Dec 03 2007
%E a(17)-a(20) from _Giovanni Resta_, Oct 28 2012
%E a(21)-a(22) from _Giovanni Resta_, Aug 12 2019