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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

%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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Last modified March 28 08:02 EDT 2024. Contains 371236 sequences. (Running on oeis4.)