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A343731
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Numbers k at which tau(k^k) reaches a record high, where tau is the number-of-divisors function A000005.
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1
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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
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OFFSET
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1,2
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LINKS
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EXAMPLE
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In the table below, asterisks indicate record high values of tau(k^k):
tau(k^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 *
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The numbers k at which those record high values occur are 0, 2, 3, 4, 5, 6, 10, 12, ...
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PROG
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(Python)
from functools import reduce
from operator import mul
from sympy import factorint
for n in range(2, 10**5):
x = reduce(mul, (n*d+1 for d in factorint(n).values()))
if x > c:
c = x
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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