A294087 Least prime p_k such that (p_k)^n has p_{k+1} as substring.

23, 11, 37, 2, 7, 5, 3, 41, 3, 13, 3, 3, 2, 2, 2, 2, 5, 5, 5, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 17, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Author

Paolo P. Lava, Feb 09 2018

Keywords

• nonn
• easy
• base

Comments

It appears that a(n) = 2 for n>153. In other words, for n>153, 3 is always a substring of 2^n. Is there any proof? See A035058.

Examples

			23^2 = 529 and 29 is the prime after 23.
11^3 = 1331 and 13 is the prime after 11.
37^4 = 1874161 and 41 is the prime after 37.
		

Crossrefs

Cf. A035058, A052073, A052075, A294088.

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:=nextprime(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);