A341213 a(n) is the smallest number m such that numbers m, m - 1, m - 2, ..., m - n + 1 have k, 2*k, 3*k, ..., n*k divisors respectively.

1, 7, 47, 1019, 154379, 59423129, 3100501318, 126544656838

Offset: 1

Author

Jaroslav Krizek, Feb 07 2021

Keywords

• nonn
• more

Comments

a(n) is the smallest number m such that tau(m) = tau(m - 1)/2 = tau(m - 2)/3 = tau(m - 3)/4 = ... = tau(m - n + 1)/n, where tau(k) = the number of divisors of k (A000005).

Corresponding values of numbers k: 1, 2, 2, 2, 4, 4, 4, 4, ...

Examples

			a(3) = 47 because 45, 46 and 47 have 6, 4, and 2 divisors respectively and there is no smaller number having this property.
		

Crossrefs

Cf. A341214 (similar sequence with primes).

Cf. A000005, A340158, A340159, A341212.

Programs

Python

def tau(n): # A000005
    d, t = 1, 0
    while d*d < n:
        if n%d == 0:
            t = t+2
        d = d+1
    if d*d == n:
        t = t+1
    return t
n, a = 1, 1 # corrected by Martin Ehrenstein, Apr 14 2021
while n > 0:
    nn, t1 = 1, tau(a)
    while nn < n and tau(a-nn) == (nn+1)*t1:
        nn = nn+1
    if nn == n:
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Feb 07 2021

Extensions

a(6) from Amiram Eldar, Feb 07 2021

a(7) from Jinyuan Wang, Feb 08 2021

a(1) corrected and extended with a(8) by Martin Ehrenstein, Apr 14 2021