A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

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

Offset

1,2

Comments

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

Links

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.

Formula

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)

Example

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.

Mathematica

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 *)

Prog

(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

Crossrefs

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

Keyword

nonn

Author

Jaroslav Krizek, Mar 01 2009

Extensions

a(1) = 1 from N. J. A. Sloane, Mar 02 2009

a(60) corrected by Klaus Brockhaus, May 26 2009

Status

approved