A302975 -id:A302975 - cp's OEIS frontend

A302974 a(n) = numerator of tau(n)^n / n^tau(n).

1, 1, 8, 81, 32, 256, 128, 16, 27, 65536, 2048, 729, 8192, 16777216, 1073741824, 152587890625, 131072, 2985984, 524288, 892616806656, 4398046511104, 1099511627776, 8388608, 281474976710656, 847288609443, 281474976710656, 18014398509481984, 1499253470328324096

Offset: 1

Jaroslav Krizek, Apr 16 2018

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

a(n) <= A302975(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) = 256.

Harvey P. Dale, Table of n, a(n) for n = 1..779

Cf. A000005, A120737, A302975, A302976.

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

Numerator[#[[2]]^#[[1]]/#[[1]]^#[[2]]]&/@Table[{n,DivisorSigma[0,n]},{n,30}] (* Harvey P. Dale, May 29 2023 *)

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

A302974 a(n) = numerator of tau(n)^n / n^tau(n).

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