A363657 -id:A363657 - cp's OEIS frontend

A217854 Product of all divisors of n, positive or negative.

-1, 4, 9, -64, 25, 1296, 49, 4096, -729, 10000, 121, 2985984, 169, 38416, 50625, -1048576, 289, 34012224, 361, 64000000, 194481, 234256, 529, 110075314176, -15625, 456976, 531441, 481890304, 841, 656100000000, 961, 1073741824, 1185921

Offset: 1

Franklin T. Adams-Watters, Oct 19 2012

sign

a(n) is negative iff n is a square.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000005, A062758, A158387, A363657, A363658.

a[n_] := (-n)^DivisorSigma[0, n]; Array[a, 33] (* Amiram Eldar, Aug 31 2023 *)

a(n) = (-n)^numdiv(n); \\ Michel Marcus, Aug 31 2023

a(n) = (-n)^tau(n) = (-n)^A000005(n).

a(n) = A158387(n) * A062758(n). - Andrew Howroyd, Aug 31 2023

A363658 Positive numbers m where A217854(m) is positive and increases to a record.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 12, 18, 20, 24, 30, 40, 42, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080

Offset: 1

Simon Jensen, Jun 13 2023

nonn

(-m)^tau(m) > 0 and (-m)^tau(m) > (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.
There are no squares in this sequence.
It appears that if n > 13, then a(n) = A067128(n). See the link.
Only a finite number of terms in A002093 can also be terms in this sequence. See the link.

			5 is a term since (-5)^tau(5) = (-5)^2 = 25 and 25 > (-k)^tau(k) for k = 1,...,4.

Simon Jensen, Table of n, a(n) for n = 1..135
Simon Jensen, On an extended divisor product summatory function

Cf. A363657, A067128, A000005, A002093, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); if (x>0, for (k=1, m-1, if (x <= (-k)^numdiv(k), return(0))); return(1)); \\ Michel Marcus, Aug 31 2023

cp's OEIS Frontend

A217854 Product of all divisors of n, positive or negative.

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