A333099 a(n) is the final term of the lexicographically first sequence of distinct positive multiples of n whose reciprocals sum to 1.

1, 12, 173880, 147211626090065500462558943962082011818610800

Offset: 1

Author

Jon E. Schoenfield, Mar 07 2020

Keywords

• nonn

Comments

a(5) is a 142549-digit number.

Let S_n be the lexicographically first sequence of distinct positive multiples of n whose reciprocals sum to 1, and let S_n(k) be the k-th term in that sequence; then for n > 1, S_n(k) = n*k iff k <= A115515(n). E.g., for n=3, S_3 = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 45, 690, 173880}, whose first A115515(3)=10 terms are 3*1, 3*2, ..., 3*10, but the 11th term (45) exceeds 33.

Examples

			For n=2, 1 - (1/2 + 1/4 + 1/6) = 1/12, i.e., 1/2 + 1/4 + 1/6 + 1/12 = 1, so a(2)=12.
For n=3, 1 - (1/3 + 1/6 + 1/9 + ... + 1/30) = 1/42.2346...;
1 - (1/3 + 1/6 + 1/9 + ... + 1/30 + 1/45) = 1/687.272727...;
1 - (1/3 + 1/6 + 1/9 + ... + 1/30 + 1/45 + 1/690) = 1/173880, i.e., 1/3 + 1/6 + 1/9 + ... + 1/30 + 1/45 + 1/690 + 1/173880 = 1, so a(3)=173880.
For n=4, the sum of reciprocals is 1/4 + 1/8 + 1/12 + ... + 1/120 + 1/800 + 1/310824 + 1/66131478848 + 1/12922318759882631742928 + 1/147211626090065500462558943962082011818610800 = 1, so a(4)=147211626090065500462558943962082011818610800.
		

Crossrefs

Cf. A115515.