A154780 Numbers k with d digits such that all digits of k and the last d+1 digits of k^2 are prime.

5, 35, 235, 335, 2335, 3335, 23335, 32335, 33335, 72335, 233335, 323335, 333335, 372335, 572335, 723335, 2333335, 2372335, 2723335, 3233335, 3323335, 3333335, 3572335, 3723335, 7233335, 7323335, 7372335, 7572335, 22372335, 23333335

Offset: 1

Author

M. F. Hasler, Jan 23 2009

Keywords

• base
• easy
• nice
• nonn

Comments

Any term with d digits is the concatenation of a prime digit and an earlier term (with d-1 digits).

The sequence is infinite since it contains subsequences b(n) = (10^n-1)/3+2 = (5,35,335,3335,...), c(n) = 23*10^n+b(n) = (235,2335,23335,...), d(n) = 3233*10^n+b(n), e(n) = 7233*10^n+b(n) etc.

Crossrefs

Subsequence of A046034; contains A153025 as a subsequence.

Programs

Mathematica

Select[Range[5,24000000,5],And@@PrimeQ[IntegerDigits[#]]&& And@@ PrimeQ[ Take[ IntegerDigits[#^2],-(IntegerLength[#]+1)]]&] (* Harvey P. Dale, Dec 31 2012 *)

PARI

last=[0]; {for( d=1,8, new=[]; forprime( p=0,9, for( k=1,#last, is_A046034((p*10^(d-1)+last[k])^2%10^(d+1)+20*10^d) & new=concat( new, p*10^(d-1)+last[k]))); print1(last=new,","))} /* for slightly more efficient code see A154779 */

Formula

For all n, a(n) == (5 mod 10).

For a(n) > 5, a(n) == 35 (mod 100).

For a(n) > 35, a(n) == 235 or 335 (mod 1000).

For a(n) > 335, a(n) == 2335 or 3335 (mod 10^4).