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a(n) is the smallest exponent > 1 such that p^a(n) ends with p, where p is the n-th prime.
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%I #34 Jul 25 2019 18:11:28

%S 5,5,2,5,11,21,21,11,21,11,11,21,6,5,21,21,11,6,21,11,21,11,21,11,21,

%T 11,101,21,51,101,101,51,101,51,11,11,21,101,101,101,51,51,51,5,101,

%U 11,51,101,101,51,101,51,26,3,21,101,51,51,101,26,101,21,5,51

%N a(n) is the smallest exponent > 1 such that p^a(n) ends with p, where p is the n-th prime.

%H Paolo P. Lava, <a href="/A273007/b273007.txt">Table of n, a(n) for n = 1..1000</a>

%e 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32;

%e 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243.

%p P:=proc(q) local d,k,n; for n from 1 to q do if isprime(n) then d:=ilog10(n)+1;

%p for k from 2 to q do if n=(n^k mod 10^d) then print(k); break; fi; od; fi; od; end: P(10^3);

%t Table[Length[NestWhileList[p #&,p^2,Mod[#,10^IntegerLength[p]]!=p&]]+1,{p,Prime[ Range[65]]}] (* _Harvey P. Dale_, Jul 25 2019 *)

%Y Cf. A051248, A115739, A273527.

%K nonn,easy,base

%O 1,1

%A _Paolo P. Lava_, May 24 2016