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A253139
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a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).
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12
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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, 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, 24, 2, 4, 12, 420, 4, 8, 2, 12, 4, 8, 2, 72, 2, 4, 12, 12, 4, 8
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OFFSET
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1,2
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COMMENTS
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A divisibility sequence (cf. Ward link and second formula).
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).
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LINKS
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FORMULA
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If n = Product_ prime(i)^e(i), then a(n) = Product_ A003418(e(i)+1).
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EXAMPLE
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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.
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MATHEMATICA
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Table[LCM@@DivisorSigma[0, Divisors[n]], {n, 100}] (* Harvey P. Dale, Sep 01 2017 *)
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 *)
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PROG
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(PARI) a(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ Michel Marcus, Jan 23 2015
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CROSSREFS
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A250270 gives range of values. A141586 lists numbers n such that a(n) divides n.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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