A339778 Numbers m such that numbers m, m + 1 and m + 2 have k, 2k and 3k divisors respectively.

61, 73, 277, 421, 458, 493, 583, 1234, 1393, 1418, 1658, 1909, 1954, 2066, 2138, 2234, 2329, 2386, 2533, 2594, 2773, 2797, 2846, 3013, 3073, 3265, 3394, 3841, 4322, 4333, 4538, 4586, 4633, 4717, 4754, 4766, 5029, 5223, 5245, 5342, 5378, 5554, 5893, 5906, 6169

Offset: 1

Author

Jaroslav Krizek, Dec 16 2020

Keywords

• nonn

Comments

Numbers m such that tau(m) = tau(m + 1) / 2 = tau(m + 2) / 3, where tau(k) = the number of divisors of k (A000005).

Corresponding values of tau(a(n)): 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 4, ...

Triplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2)] = [tau(a(n)), 2*tau(a(n)), 3*tau(a(n))]: [2, 4, 6], [2, 4, 6], [2, 4, 6], [2, 4, 6], [4, 8, 12], [4, 8, 12], [4, 8, 12], [4, 8, 12], [4, 8, 12], ...

Examples

			tau(61) = 2, tau(62) = 4, tau(63) = 6.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A100363.

Subsequence of A063446.

Programs

Magma

[m: m in [1..10^5] | #Divisors(m) eq #Divisors(m + 1) / 2 and #Divisors(m) eq #Divisors(m + 2) / 3]

Mathematica

Select[Range[6000], Equal @@ (DivisorSigma[0, # + {0, 1, 2}]/{1, 2, 3}) &] (* Amiram Eldar, Dec 16 2020 *)

PARI

isok(m) = my(nb = numdiv(m)); (numdiv(m+1) == 2*nb) && (numdiv(m+2) == 3*nb); \\ Michel Marcus, Dec 18 2020