%I #24 Jun 12 2021 13:49:05
%S 0,2,3,4,6,10,12,18,20,24,30,42,60,78,84,90,114,120,140,150,156,168,
%T 180,210,330,390,420,510,546,570,630,660,780,840,990,1020,1050,1092,
%U 1140,1170,1260,1530,1540,1560,1680,1848,1890,1980,2100,2280,2310,2730,3570
%N Numbers k at which tau(k^k) reaches a record high, where tau is the number-of-divisors function A000005.
%H Jon E. Schoenfield, <a href="/A343731/b343731.txt">Table of n, a(n) for n = 1..10000</a> (first 510 terms from Chai Wah Wu)
%e In the table below, asterisks indicate record high values of tau(k^k):
%e tau(k^k) =
%e k k^k = A000312(k) A062319(k)
%e -- ---------------- ----------
%e 0 1 1 *
%e 1 1 1
%e 2 4 3 *
%e 3 27 4 *
%e 4 256 9 *
%e 5 3125 6
%e 6 46656 49 *
%e 7 823543 8
%e 8 16777216 25
%e 9 387420489 19
%e 10 10000000000 121 *
%e 11 285311670611 12
%e 12 8916100448256 325 *
%e .
%e The numbers k at which those record high values occur are 0, 2, 3, 4, 5, 6, 10, 12, ...
%o (Python)
%o from functools import reduce
%o from operator import mul
%o from sympy import factorint
%o c, A343731_list = 0, [0]
%o for n in range(2,10**5):
%o x = reduce(mul,(n*d+1 for d in factorint(n).values()))
%o if x > c:
%o c = x
%o A343731_list.append(n) # _Chai Wah Wu_, Jun 03 2021
%Y Cf. A000005, A000312, A062319.
%K nonn
%O 1,2
%A _Jon E. Schoenfield_, Jun 01 2021
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