A360324 Numbers k such that k divides Sum_{i=1..k} 10^(1 + floor(log_10(p(i)))) - 1 - p(i), where p(i) is the i-th prime number.

1, 13, 313, 1359, 245895, 131186351, 468729047, 1830140937

Offset: 1

Author

Ctibor O. Zizka, Feb 04 2023

Keywords

• nonn
• more

Comments

Alternative Name : The arithmetic mean of the first k 9's complements of primes is an integer.

Examples

			k = 13: first 13 prime numbers are {2,3,5,7,11,13,17,19,23,29,31,37,41}, their 9's complements are {7,6,4,2,88,86,82,80,76,70,68,62,58} and (7 + 6 + ... + 62 + 58) / 13 = 53, thus 13 is a term.
		

Crossrefs

Cf. A000040, A045345, A055642, A061601.

Programs

Mathematica

s = 0; p = 2; pow = 10; seq = {}; Do[s += pow - 1 - p; If[Divisible[s, k], AppendTo[seq, k]]; p = NextPrime[p]; If[p > pow, pow *= 10], {k, 1, 250000}]; seq (* Amiram Eldar, Feb 04 2023 *)

Formula

k: (Sum_{i=1..k} 10^(1 + floor(log_10(p(i)))) - 1 - p(i)) / k = c, c an integer.

Extensions

a(5)-a(8) from Amiram Eldar, Feb 04 2023