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A343731
Numbers k at which tau(k^k) reaches a record high, where tau is the number-of-divisors function A000005.
1
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
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jun 01 2021
STATUS
approved