A160432 Primes of the form 3*10^(2*n) + 3*10^n + 1.

7, 331, 300030001, 3000000003000000001

Offset: 1

Author

Giacomo Fecondo, May 13 2009

Keywords

• nonn

Comments

Primes of the form (x^3-y^3)/(x-y) with x = y+1 (which gives A002407) and also y=10^k for some k.

These prime numbers (differences of consecutive cubes: A002407), for k>0, have only three digits different from zero. The first is 3, the middle digit is 3 and the final digit is 1. The other 2(k-1) digits are value 0.

If k=6*i or k=6*i-1 the number is always divisible by 7. [Giacomo Fecondo, May 22 2010]

Examples

			a(1) = 7 = (10^0+1)^3 -(10^0)^3 , 2^3-1^3.
a(2) = 331 =(10^1+1)^3 -(10^1)^3, 11^3-10^3.
a(3) = 300030001 = (10^4+1)^3 - (10^4)^3, 10001^3-10000^3.
a(1)= 3t(t+1)+1 with t=10^0; a(2)= 3t(t+1)+1 with t=10^1; a(3)= 3t(t+1)+1 with t=10^4.
For k=102 (k=6*17) the number (10^102+1)^3-(10^102)^3 is divisible by 7; for k=101 (k=6*17-1) the number (10^101+1)^3-(10^101)^3 is divisible by 7. [_Giacomo Fecondo_, May 22 2010]
		

Crossrefs

Cf. A002407, A003215.

Programs

Magma

[a: n in [0..30] | IsPrime(a) where a is 3*10^(2*n) + 3*10^n + 1]; // Vincenzo Librandi, Jan 28 2013

Mathematica

Select[Table[3*10^(2 n) + 3*10^n + 1, {n, 0, 1000}], PrimeQ] (* Vincenzo Librandi, Jan 28 2013 *)

PARI

A160432(n,print_all=0,Start=0,Limit=9e9)={for(k=Start,Limit,ispseudoprime(p=3*100^k+3*10^k+1) & !(print_all & print1(p",")) & !n-- & return(p))} \\ - M. F. Hasler, Jan 28 2013

Extensions

New name from Vincenzo Librandi, Jan 28 2013