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

Author

Jon E. Schoenfield, Jun 01 2021

Keywords

• nonn

Examples

			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, ...
		

Links

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

Crossrefs

Cf. A000005, A000312, A062319.

Programs

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 *)

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