A342439 Let S(n,k) denote the set of primes < 10^n which are the sum of k consecutive primes, and let K = maximum k >= 2 such that S(n,k) is nonempty; then a(n) = max S(n,K).

5, 41, 953, 9521, 92951, 997651, 9964597, 99819619, 999715711, 9999419621, 99987684473, 999973156643, 9999946325147, 99999863884699, 999999149973119, 9999994503821977, 99999999469565483, 999999988375776737, 9999999776402081701

Offset: 1

Author

Bernard Schott, Mar 12 2021

Keywords

• nonn
• base
• more

Comments

Inspired by the 50th problem of Project Euler (see link).

There must be at least two consecutive primes in the sum.

The corresponding number K of consecutive primes to get this largest prime is A342440(n) and the first prime of these A342440(n) consecutive primes is A342453(n).

It can happen that the sums of K = A342440(n) consecutive primes give two (or more) distinct n-digit primes. In that case, a(n) is the greatest of these primes. Martin Ehrenstein proved that there are only two such cases when 1 <= n <= 19, for n = 7 and n = 15 (see corresponding examples).

Solutions and Python program are proposed in Dreamshire and Archive.today links. - Daniel Suteu, Mar 12 2021

Examples

			a(1) = 5 = 2+3.
a(2) = 41 = 2 + 3 + 5 + 7 + 11 + 13; note that 97 = 29 + 31 + 37 is prime, sum of 3 consecutive primes, but 41 is obtained by adding 6 consecutive primes, so, 97 is not a term.
A342440(7) = 1587, and there exist two 7-digit primes that are sum of 1587 consecutive primes; as 9951191 = 5+...+13399 < 9964597 = 7+...+13411 hence a(7) = 9964597.
A342440(15) = 10695879 , and there exist two 15-digit primes that are sum of 10695879 consecutive primes; as 999998764608469 = 7+...+192682309 < 999999149973119 = 13+...+192682337, hence a(15) = 999999149973119.
		

Links

• Dreamshire, Project Euler 50 Solution.
• Archive.today, trizen / experimental-projects.
• Project Euler, Problem 50: Consecutive prime sum.

Crossrefs

Cf. A034962, A034965, A082246, A082251, A127340, A127341, A215991, A067377.

Cf. A342440, A342443, A342444, A342453, A342454.

Extensions

Name improved by N. J. A. Sloane, Mar 12 2021

a(4)-a(17) from Daniel Suteu, Mar 12 2021

a(18)-a(19) from Martin Ehrenstein, Mar 13 2021

a(7) and a(15) corrected by Martin Ehrenstein, Mar 15 2021