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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.

%I A341213 #23 Apr 15 2021 01:21:47
%S A341213 1,7,47,1019,154379,59423129,3100501318,126544656838
%N 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.
%C A341213 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 A341213 Corresponding values of numbers k: 1, 2, 2, 2, 4, 4, 4, 4, ...
%e A341213 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 A341213 (Python)
%o A341213 def tau(n): # A000005
%o A341213     d, t = 1, 0
%o A341213     while d*d < n:
%o A341213         if n%d == 0:
%o A341213             t = t+2
%o A341213         d = d+1
%o A341213     if d*d == n:
%o A341213         t = t+1
%o A341213     return t
%o A341213 n, a = 1, 1 # corrected by _Martin Ehrenstein_, Apr 14 2021
%o A341213 while n > 0:
%o A341213     nn, t1 = 1, tau(a)
%o A341213     while nn < n and tau(a-nn) == (nn+1)*t1:
%o A341213         nn = nn+1
%o A341213     if nn == n:
%o A341213         print(n,a)
%o A341213         n = n+1
%o A341213     a = a+1 # _A.H.M. Smeets_, Feb 07 2021
%Y A341213 Cf. A341214 (similar sequence with primes).
%Y A341213 Cf. A000005, A340158, A340159, A341212.
%K A341213 nonn,more
%O A341213 1,2
%A A341213 _Jaroslav Krizek_, Feb 07 2021
%E A341213 a(6) from _Amiram Eldar_, Feb 07 2021
%E A341213 a(7) from _Jinyuan Wang_, Feb 08 2021
%E A341213 a(1) corrected and extended with a(8) by _Martin Ehrenstein_, Apr 14 2021