A321613 Partial products of the unitary totient function (A047994): a(n) = Product_{k=1..n} uphi(k).

1, 1, 2, 6, 24, 48, 288, 2016, 16128, 64512, 645120, 3870720, 46448640, 278691840, 2229534720, 33443020800, 535088332800, 4280706662400, 77052719923200, 924632639078400, 11095591668940800, 110955916689408000, 2441030167166976000, 34174422340337664000

Offset: 1

Author

Amiram Eldar, Dec 19 2018

Keywords

• nonn

Comments

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = ugcd(i,j) for 1 <= i,j <= n, where ugcd(i,j) in the greatest common unitary divisor of i and j (A165430).

The unitary version of A001088.

Examples

			a(4) = uphi(1) * uphi(2) * uphi(3) * uphi(4) = 1 * 1 * 2 * 3 = 6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..481
• H. Jager, The unitary analogues of some identities for certain arithmetical functions, Nederl. Akad. Wetensch. Proc. Ser. A, Vol. 64 (1961), pp. 508-515.

Crossrefs

Cf. A001088, A047994, A165430, A177754.

Programs

Mathematica

uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); FoldList[ Times, uphi /@ Range[50]]

PARI

uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994
a(n) = prod(k=1, n, uphi(k)); \\ Michel Marcus, Dec 19 2018