A338968 a(n) is the largest prime whose decimal expansion consists of the concatenation of a 1-digit prime, a 2-digit prime, a 3-digit prime, ..., and an n-digit prime.

7, 797, 797977, 7979979941, 797997997399817, 797997997399991999371, 7979979973999919999839999901, 797997997399991999983999999199999131, 797997997399991999983999999199999989999997639, 7979979973999919999839999991999999899999999379999997871

Offset: 1

Author

Bernard Schott, Dec 21 2020

Keywords

• nonn
• base

Comments

It is a plausible conjecture that a(n) always exists and begins with 7.

The similar smallest primes are in A215641.

If a(n) exists, it has A000217(n) = n*(n+1)/2 digits.

a(1) = 7 = A003618(1) and a(2) = 797 is the concatenation of 7 = A003618(1) and 97 = A003618(2) that are respectively the largest 1-digit prime and 2-digit prime.

Conjecture: for n >= 3, a(n) is the concatenation of the largest k-digit primes with 1 <= k <= n-1: A003618(1)/A003618(2)/.../A003618(n-1) but the last concatenated prime with n digits is always < A003618(n). This conjecture has been checked by Daniel Suteu until a(360), a prime with 64980 digits.

Examples

			a(3) = 797977 is the largest prime formed from the concatenation of a single-digit, a double-digit, a triple-digit prime, i.e., 7, 97, 977.
a(4) = 7979979941 is the largest prime formed from the concatenation of a single-digit, a double-digit, a triple-digit, and a quadruple-digit prime, i.e., 7, 97, 997, 9941.
		

Crossrefs

Cf. A000217, A003618, A215641.

Subsequence of A195302.

Cf. A339978 (with concatenated squares), A340115 (with concatenated cubes).

Extensions

More terms from David A. Corneth, Dec 21 2020