A217307 Minimal natural number (in decimal representation) with n prime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

1, 2, 16, 17, 115, 121, 509, 821, 3251, 4721, 5749, 22760, 25301, 41673, 142950, 173819, 291714, 920561, 1222716, 2041709, 4450031, 8559017, 9350687, 14295199, 31150219, 50568439, 71502954, 100066398, 218051538, 353979075, 500526787, 702815371, 1512442643

Offset: 0

Author

Hieronymus Fischer, Nov 22 2012

Keywords

• nonn
• base

Comments

The sequence is well-defined in that for each n the set of numbers with n prime substrings is not empty. Proof: Define m(0):=1, m(1):=2 and m(n+1):=7*m(n)+2 for n>0. This results in m(n)=2*sum_{j=0..n-1} 7^j = (7^n - 1)/3 or m(n)=1, 2, 22, 222, 2222, 22222,…, (in base-7) for n=0,1,2,3,…. Evidently, for n>0 m(n) has n 2’s and these are the only prime substrings in base-7 representation. This is why every substring of m(n) with more than one digit is a product of two integers > 1 (by definition) and can therefore not be prime number.

No term is divisible by 7.

Examples

			a(1) = 2 = 2_7, since 2 is the least number with 1 prime substring in base-7 representation.
a(2) = 16 = 22_7, since 16 is the least number with 2 prime substrings in base-7 representation (2 times 2_7=2).
a(3) = 17 = 23_7, since 17 is the least number with 3 prime substrings in base-7 representation (2_7, 3_7, and 23_7).
a(5) = 121 = 232_7, since 121 is the least number with 5 prime substrings in base-7 representation (2 times 2_7, 3_7, 23_7=17, and 32_7=23).
a(6) = 509 = 1325_7, since 509 is the least number with 6 prime substrings in base-7 representation (2_7, 3_7, 5_7, 25_7=19, 32_7=23, and 1325_7=509).
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..40

Crossrefs

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

Cf. A035244, A079397, A213300-A213321.

Cf. A217302-A217309.

Formula

a(n) > 7^floor(sqrt(8*n-7)-1)/2), for n>0.

a(n) <= (7^n - 1)/3, n>0.

a(n+1) <= 7*a(n) + 2.