A302974 -id:A302974 - cp's OEIS frontend

A302975 a(n) = denominator of tau(n)^n / n^tau(n).

1, 1, 9, 64, 25, 81, 49, 1, 1, 625, 121, 1, 169, 2401, 50625, 1048576, 289, 1, 361, 15625, 194481, 14641, 529, 6561, 15625, 28561, 531441, 117649, 841, 2562890625, 961, 1, 1185921, 83521, 1500625, 262144, 1369, 130321, 2313441, 390625, 1681, 37822859361, 1849

Offset: 1

Jaroslav Krizek, Apr 16 2018

tau(n) = the number of the divisors of n (A000005).
Conjecture: all terms are squares.
a(n) >= A302974(n) only for numbers n = 1, 2 and 3.

			For n = 6; tau(6)^6 / 6^tau(6) = 4^6 / 6^4 = 256 / 81; a(6) = 81.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A120737, A302974, A302976.

[Denominator((NumberOfDivisors(n)^n) / (n^NumberOfDivisors(n))): n in[1..100]];

Denominator[#[[2]]^#[[1]]/#[[1]]^#[[2]]]&/@Table[{n,DivisorSigma[0,n]},{n,50}] (* Harvey P. Dale, Sep 15 2019 *)

a(p) = p^2 for p = prime.

a(A120737(n)) = 1.

A302976 a(n) = tau(n)^n mod n^tau(n).

Original entry on oeis.org

0, 0, 8, 17, 7, 208, 30, 0, 0, 8576, 112, 0, 80, 22864, 36199, 159681, 155, 0, 116, 40062976, 83791, 142928, 255, 26138902528, 68, 302656, 362152, 454885376, 60, 544999124224, 374, 0, 226279, 629152, 399674, 27234498115233, 76, 956704, 956539, 3361080344576

Offset: 1

Jaroslav Krizek, Apr 16 2018

nonn

tau(n) = the number of the divisors of n (A000005).

tau(n)^n > n^tau(n) for all n > 3.

			For n = 8; a(8) = 0 because tau(8)^8 mod 8^tau(8) = 4^8 mod 8^4 = 65536 mod 4096 = 0.

Cf. A000005, A120737, A302974, A302975.

[(NumberOfDivisors(n)^n) mod (n^NumberOfDivisors(n)): n in[1..100]];

PowerMod[#[[2]],#[[1]],#[[1]]^#[[2]]]&/@Table[{n,DivisorSigma[0,n]},{n,40}] (* Harvey P. Dale, Jan 08 2023 *)

a(n) = A000005(n)^n mod n^A000005(n) = A302974(n) mod A302975(n).

a(A120737(n)) = 0.

cp's OEIS Frontend

A302975 a(n) = denominator of tau(n)^n / n^tau(n).

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