A343731 Numbers k at which tau(k^k) reaches a record high, where tau is the number-of-divisors function A000005.

0, 2, 3, 4, 6, 10, 12, 18, 20, 24, 30, 42, 60, 78, 84, 90, 114, 120, 140, 150, 156, 168, 180, 210, 330, 390, 420, 510, 546, 570, 630, 660, 780, 840, 990, 1020, 1050, 1092, 1140, 1170, 1260, 1530, 1540, 1560, 1680, 1848, 1890, 1980, 2100, 2280, 2310, 2730, 3570

Offset

1,2

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 510 terms from Chai Wah Wu)

Example

In the table below, asterisks indicate record high values of tau(k^k):
                        tau(k^k) =
   k  k^k = A000312(k)  A062319(k)
  --  ----------------  ----------
   0                 1           1 *
   1                 1           1
   2                 4           3 *
   3                27           4 *
   4               256           9 *
   5              3125           6
   6             46656          49 *
   7            823543           8
   8          16777216          25
   9         387420489          19
  10       10000000000         121 *
  11      285311670611          12
  12     8916100448256         325 *
.
The numbers k at which those record high values occur are 0, 2, 3, 4, 5, 6, 10, 12, ...

Mathematica

Join[{0}, DeleteDuplicates[Table[{n, DivisorSigma[0, n^n]}, {n, 2, 3600}], GreaterEqual[ #1[[2]], #2[[2]]]&][[;; , 1]]] (* Harvey P. Dale, Jul 21 2024 *)

Prog

(Python)
from functools import reduce
from operator import mul
from sympy import factorint
c, A343731_list = 0, [0]
for n in range(2, 10**5):
    x = reduce(mul, (n*d+1 for d in factorint(n).values()))
    if x > c:
        c = x
        A343731_list.append(n) # Chai Wah Wu, Jun 03 2021

Crossrefs

Cf. A000005, A000312, A062319.

Sequence in context: A069744 A229362 A249685 * A181312 A373320 A330006

Adjacent sequences: A343728 A343729 A343730 * A343732 A343733 A343734

Keyword

nonn

Author

Jon E. Schoenfield, Jun 01 2021

Status

approved