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Numbers k such that A101337(k)/k is an integer.
8

%I #44 Nov 05 2023 15:23:35

%S 1,2,3,4,5,6,7,8,9,153,370,371,407,459,1634,8208,9474,13598,48495,

%T 54748,92727,93084,119564,174961,306979,548834,1741725,3194922,

%U 4210818,9800817,9926315,12720569,24678050,24678051,88593477,144688641,146511208

%N Numbers k such that A101337(k)/k is an integer.

%C A005188 is a subsequence of this sequence.

%C Sequence is finite. In particular, a(n) < 10^60. If k >= 10^60, then A101337(k) < k. - _Chai Wah Wu_, Feb 26 2019

%H Giovanni Resta, <a href="/A306360/b306360.txt">Table of n, a(n) for n = 1..109</a> (full sequence; first 56 terms from Chai Wah Wu)

%H Barry Fagin, <a href="https://doi.org/10.3390/info10070232">Idempotent Factorizations of Square-free Integers</a>, Preprints (2019).

%e For k = 1, (1^1)/1 = 1;

%e for k = 459, (4^3 + 5^3 + 9^3) / 459 = 2.

%t Select[Range[10^6], IntegerQ[Total[IntegerDigits[#]^IntegerLength[#]]/#] &] (* _Michael De Vlieger_, Aug 01 2019 *)

%o (PARI) isok(n) = frac(A101337(n)/n) == 0; \\ _Michel Marcus_, Feb 11 2019

%o (PARI) select( is(n)=!(A101337(n)%n), [0..999]) \\ _M. F. Hasler_, Nov 17 2019

%o (Python)

%o A306360_list, k = [], 1

%o while k < 10**9:

%o s = str(k)

%o l, c = len(s), 0

%o for i in range(l):

%o c = (c + int(s[i])**l) % k

%o if c == 0:

%o A306360_list.append(k)

%o k += 1 # _Chai Wah Wu_, Feb 26 2019

%Y Cf. A005188, A101337, A306354, A306361.

%K nonn,base,fini,full

%O 1,2

%A _Ctibor O. Zizka_, Feb 10 2019

%E a(22)-a(37) from _Daniel Suteu_, Feb 10 2019