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%I #25 Sep 11 2020 04:15:23
%S 1,2,2,6,2,4,2,12,6,4,2,12,2,4,4,60,2,12,2,12,4,4,2,24,6,4,12,12,2,8,
%T 2,60,4,4,4,36,2,4,4,24,2,8,2,12,12,4,2,120,6,12,4,12,2,24,4,24,4,4,2,
%U 24,2,4,12,420,4,8,2,12,4,8,2,72,2,4,12,12,4,8
%N a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).
%C A divisibility sequence (cf. Ward link and second formula).
%C a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
%H Antti Karttunen, <a href="/A253139/b253139.txt">Table of n, a(n) for n = 1..10000</a>
%H Morgan Ward, <a href="http://dx.doi.org/10.1090/S0002-9904-1939-06980-2">A note on divisibility sequences</a>, Bull. Amer. Math. Soc., 45 (1939), 334-336.
%F If n = Product_ prime(i)^e(i), then a(n) = Product_ A003418(e(i)+1).
%F a(n) = Product_{d|n} A253141(d).
%e The divisors of 20 are 1, 2, 4, 5, 10 and 20, which have 1, 2, 3, 2, 4 and 6 divisors respectively. The least common multiple of 1, 2, 3, 2, 4 and 6 is 12; therefore, a(20) = 12.
%t Table[LCM@@DivisorSigma[0,Divisors[n]],{n,100}] (* _Harvey P. Dale_, Sep 01 2017 *)
%t lcm[n_] := lcm[n] = LCM @@ Range[n]; a[1] = 1; a[n_] := Times @@ (lcm [Last[#] + 1] & /@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 11 2020 *)
%o (PARI) a(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ _Michel Marcus_, Jan 23 2015
%Y A250270 gives range of values. A141586 lists numbers n such that a(n) divides n.
%Y Cf. A003418, A253141.
%K nonn,easy,mult
%O 1,2
%A _Matthew Vandermast_, Dec 27 2014
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