A291465 a(n) is the least m >= n for which the complete bipartite graph K_{m,n} has a prime labeling.

1, 2, 4, 9, 14, 25, 36, 45, 52, 61, 62, 89, 90, 95, 98, 123, 140, 155, 162, 171, 172, 177, 216, 217, 226, 243, 244, 255, 264, 283, 318, 321, 340, 345, 374, 383, 384, 395, 400, 403, 422, 449, 456, 465, 478, 531, 546, 551, 552, 557, 562, 567, 594, 599, 604, 605

Offset: 1

Author

Jonathan Sondow, Aug 24 2017

Keywords

• nonn

Comments

A prime labeling of K_{m,n} is a pair of sets A and B whose union is {1,2,...,m+n} such that |A| = m, |B| = n, and gcd(a,b) = 1 for all a in A and b in B. For an equivalent definition, the data above, and the formula below involving R_{n-1}, see Berliner, Dean, Hook, Marr, Mbirika (2016) Section 3.2.

Examples

			A = {1,3} and B = {2,4} is a prime labeling of K_{2,2}, so a(2) = 2.
		

Links

• Paul Tabatabai, Table of n, a(n) for n = 1..75
• Adam H. Berliner, N. Dean, J. Hook, A. Marr, and A. Mbirika, Coprime and prime labelings of graphs, arXiv:1604.07698 [math.CO], 2016; Journal of Integer Sequences, Vol. 19 (2016), #16.5.8.

Crossrefs

Cf. A104272, A213273, A213806, A284875.

Formula

n+1 <= a(n) <= R_{n-1} - n for n > 2, where R_{n-1} is a Ramanujan prime A104272.

Extensions

a(14) onward from Paul Tabatabai, Apr 29 2019