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

1,2

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, ...

Links

Table of n, a(n) for n=1..8.

Example

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

Prog

(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

Crossrefs

Cf. A341214 (similar sequence with primes).

Cf. A000005, A340158, A340159, A341212.

Sequence in context: A368295 A268063 A015097 * A201176 A013400 A013399

Adjacent sequences: A341210 A341211 A341212 * A341214 A341215 A341216

Keyword

nonn,more

Author

Jaroslav Krizek, 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

Status

approved