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A157488
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a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.
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3
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1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 72, 13, 14, 15, 128, 17, 108, 19, 200, 21, 22, 23, 144, 125, 26, 81, 392, 29, 30, 31, 64, 33, 34, 35, 46656, 37, 38, 39, 400, 41, 42, 43, 968, 675, 46, 47, 3456, 343, 500, 51, 1352, 53, 324, 55, 784, 57, 58, 59, 1800, 61, 62, 1323, 4096
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OFFSET
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1,2
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COMMENTS
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The exponential divisors of a number n = Product p(i)^e(i) are all numbers of the form Product p(i)^s(i) where s(i) divides e(i) for all i.
Not multiplicative: a(3)=3 (e-divisor 3^1), a(4)=8 (e-divisors 2^1 and 2^2), but a(12)=72 (e-divisors 3*2 and 3*2^2) <> a(3)*a(4). - R. J. Mathar, Apr 14 2011
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LINKS
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Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, A note on exponential divisors and related arithmetic functions, Scientia Magna, Vol. 1, No. 1 (2005), pp. 97-101.
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FORMULA
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a(1) = 1, a(p) = p, a(p*q) = p*q, a(p*q...*z) = pq...z, a(p^k) = p^(A000203(k)), for p, q, ..., z distinct primes and k > 1 an integer.
From Amiram Eldar, Jun 03 2020: (Start)
If n = Product_{i} p_i^e_i then a(n) = Product_{i} p_i^(sigma(e_i) * d_exp(n) / d(e_i)), where d_exp(n) = Product_{i} d(e_i) is the number of exponential divisors of n (A049419), d(e) and sigma(e) are the number of divisors (A000005) of e and their sum (A000203).
a(n) <= A007955(n) with equality if and only if n is noncomposite. (End)
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EXAMPLE
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For n = 16 = 2^4 = the a(16) = 2^(A000203(4)) = 2^7 = 128. e-divisors of number 16 is 2, 4, 16, their product is 128.
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MATHEMATICA
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f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] :=(Times @@ (f @@@ (fct = FactorInteger[n])))^(Times @@ DivisorSigma[0, Last /@ fct]); Array[a, 100] (* Amiram Eldar, Jun 03 2020 *)
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PROG
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(Magma) cat [ &*[ d: d in Divisors(n) | forall(t) { p: p in P | v gt 0 and e mod v eq 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..64] ]; // Klaus Brockhaus, May 26 2009
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CROSSREFS
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Cf. A000005, A000027, A000040, A000203, A006881, A000961, A049419, A051377, A120944, A322791.
Similar sequences: A007955, A061537, A274029.
Sequence in context: A242480 A175068 A355264 * A188385 A332793 A102631
Adjacent sequences: A157485 A157486 A157487 * A157489 A157490 A157491
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KEYWORD
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nonn
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AUTHOR
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Jaroslav Krizek, Mar 01 2009
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EXTENSIONS
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a(1) = 1 from N. J. A. Sloane, Mar 02 2009
a(60) corrected by Klaus Brockhaus, May 26 2009
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STATUS
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approved
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