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

1,1

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.

Links

Table of n, a(n) for n=1..30.

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).

Mathematica

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

Prog

(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 */

Crossrefs

Subsequence of A046034; contains A153025 as a subsequence.

Sequence in context: A002074 A187444 A166176 * A007995 A091928 A305739

Adjacent sequences: A154777 A154778 A154779 * A154781 A154782 A154783

Keyword

base,easy,nice,nonn

Author

M. F. Hasler, Jan 23 2009

Status

approved