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A077497 Primes of the form 2^r*5^s + 1.

%I A077497 #33 Sep 19 2022 19:32:51
%S A077497 2,3,5,11,17,41,101,251,257,401,641,1601,4001,16001,25601,40961,62501,
%T A077497 65537,160001,163841,16384001,26214401,40960001,62500001,104857601,
%U A077497 167772161,256000001,409600001,655360001,2441406251,2500000001,4194304001,10485760001
%N A077497 Primes of the form 2^r*5^s + 1.
%C A077497 These are also the prime numbers p for which there is an integer solution x to the equation p*x = p*10^p + x, or equivalently, the prime numbers p for which (p*10^p)/(p-1) is an integer. - _Vicente Izquierdo Gomez_, Feb 20 2013
%C A077497 For n > 2, all terms are congruent to 5 (mod 6). - _Muniru A Asiru_, Sep 03 2017
%H A077497 Ray Chandler, <a href="/A077497/b077497.txt">Table of n, a(n) for n = 1..3150</a>
%e A077497 101 is in the sequence, since 101 = 2^2*5^2 + 1 and 101 is prime.
%t A077497 Do[p=Prime[k];s=FindInstance[p x == p 10^p+x,x,Integers];If[s!={},Print[p]],{k,10000}] (* _Vicente Izquierdo Gomez_, Feb 20 2013 *)
%o A077497 (PARI) list(lim)=my(v=List(),t);for(r=0,log(lim)\log(5),t=5^r;while(t<=lim,if(isprime(t+1),listput(v,t+1)); t<<=1)); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jan 29 2013
%o A077497 (GAP)
%o A077497 K:=10^7;; # to get all terms <= K.
%o A077497 A:=Filtered(Filtered([1..K],i-> i mod 6=5),IsPrime);;
%o A077497 B:=List(A,i->Factors(i-1));;
%o A077497 C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,5]  then Add(C,Position(B,i)); fi; od;
%o A077497 A077497:=Concatenation([2,3],List(C,i->A[i])); # _Muniru A Asiru_, Sep 03 2017
%Y A077497 Cf. A005109, A077497, A077498, A077500, A003592, A077313, A019434.
%K A077497 nonn
%O A077497 1,1
%A A077497 _Amarnath Murthy_, Nov 07 2002
%E A077497 Corrected and extended by _Reinhard Zumkeller_, Nov 19 2002
%E A077497 More terms from _Ray Chandler_, Aug 02 2003