A359261 a(n) is the least term of A359260 whose number of divisors is n.

1, 3, 49, 15, 923521, 1519, 88245939632761, 3913, 1117249, 3131659711, 4345096786921664259621718196367601, 238483, 9024585590445680759701490904755712009585829774768244676951841, 2772760313554466311, 198528059518891985825881, 32748812641

Offset: 1

Author

Amiram Eldar, Dec 23 2022

Keywords

• nonn

Comments

a(n) is the least number m whose number of divisors is A000005(m) = n such that the arithmetic mean of the first k divisors of m is an integer for all k in 1..n.

a(17) = 4084081^16 = 5.991...*10^105 is too large to include in the data section.

a(n) exists for all n >= 1. For n > 1, consider a prime p of the form m*lcm(1,2,...n-1) + 1, with m >= 1. Such a prime exists by Dirichlet's theorem on arithmetic progressions. Then, p^(n-1) has n divisors, and p^k == 1 (mod lcm(1..n-1)) for k = 0..(n-1). Therefore, Sum_{k=0..n-1} p^k == k (mod lcm(1,2,...n-1)), or equivalently, Sum_{k=0..n-1} p^k is divisible by k for k = 0..(n-1). Thus, p^(n-1) is in A359260.

Examples

			a(3) = 49 since 49 is the least number with 3 divisors in A359260. Its divisors are {1, 7, 49}, 1/1 = 1, (1+7)/2 = 4, and (1+7+49)/3 = 19 are all integers.
		

Crossrefs

Cf. A000005, A003418, A359260, A359262.

Similar sequence: A334421.