A294088 Least prime p_k such that (p_k)^n has p_{k-1} as substring.

3701, 3, 43, 3, 3, 3, 5, 5, 7, 11, 11, 3, 3, 5, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 2

Author

Paolo P. Lava, Feb 09 2018

Keywords

• nonn
• easy
• base

Comments

It appears that a(n) = 3 for n>59. In other words, for n>59, 2 is always a substring of 3^n. Is there any proof? See A131625.

Examples

			3701^2 = 13697401 and 3697 is the prime before 3701.
3^3 = 27 and 2 is the prime before 3.
43^4 = 3418801 and 41 is the prime before 43.
		

Crossrefs

Cf. A052073, A052075, A131625, A294087.

Programs

Maple

P:=proc(q) local a,b,h,k,n,ok; for h from 2 to q do ok:=1; for n from 1 to q do
if ok=1 then a:=ithprime(n); b:=prevprime(a); for k from 1 to ilog10(a^h)-ilog10(b)+1 do
if b=trunc(a^h/10^(k-1)) mod 10^(ilog10(b)+1) then print(a); ok:=0; break;
fi; od; fi; od; od; end: P(10^6);