A341213 - OEIS

%I #23 Apr 15 2021 01:21:47

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

%N 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.

%C 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).

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

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

%o (Python)

%o def tau(n): # A000005

%o     d, t = 1, 0

%o     while d*d < n:

%o         if n%d == 0:

%o             t = t+2

%o         d = d+1

%o     if d*d == n:

%o         t = t+1

%o     return t

%o n, a = 1, 1 # corrected by _Martin Ehrenstein_, Apr 14 2021

%o while n > 0:

%o     nn, t1 = 1, tau(a)

%o     while nn < n and tau(a-nn) == (nn+1)*t1:

%o         nn = nn+1

%o     if nn == n:

%o         print(n,a)

%o         n = n+1

%o     a = a+1 # _A.H.M. Smeets_, Feb 07 2021

%Y Cf. A341214 (similar sequence with primes).

%Y Cf. A000005, A340158, A340159, A341212.

%K nonn,more

%O 1,2

%A _Jaroslav Krizek_, Feb 07 2021

%E a(6) from _Amiram Eldar_, Feb 07 2021

%E a(7) from _Jinyuan Wang_, Feb 08 2021

%E a(1) corrected and extended with a(8) by _Martin Ehrenstein_, Apr 14 2021