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A306360 Numbers k such that A101337(k)/k is an integer.

%I A306360 #44 Nov 05 2023 15:23:35
%S A306360 1,2,3,4,5,6,7,8,9,153,370,371,407,459,1634,8208,9474,13598,48495,
%T A306360 54748,92727,93084,119564,174961,306979,548834,1741725,3194922,
%U A306360 4210818,9800817,9926315,12720569,24678050,24678051,88593477,144688641,146511208
%N A306360 Numbers k such that A101337(k)/k is an integer.
%C A306360 A005188 is a subsequence of this sequence.
%C A306360 Sequence is finite. In particular, a(n) < 10^60. If k >= 10^60, then A101337(k) < k. - _Chai Wah Wu_, Feb 26 2019
%H A306360 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 A306360 Barry Fagin, <a href="https://doi.org/10.3390/info10070232">Idempotent Factorizations of Square-free Integers</a>, Preprints (2019).
%e A306360 For k = 1, (1^1)/1 = 1;
%e A306360 for k = 459, (4^3 + 5^3 + 9^3) / 459 = 2.
%t A306360 Select[Range[10^6], IntegerQ[Total[IntegerDigits[#]^IntegerLength[#]]/#] &] (* _Michael De Vlieger_, Aug 01 2019 *)
%o A306360 (PARI) isok(n) = frac(A101337(n)/n) == 0; \\ _Michel Marcus_, Feb 11 2019
%o A306360 (PARI) select( is(n)=!(A101337(n)%n), [0..999]) \\ _M. F. Hasler_, Nov 17 2019
%o A306360 (Python)
%o A306360 A306360_list, k = [], 1
%o A306360 while k < 10**9:
%o A306360     s = str(k)
%o A306360     l, c = len(s), 0
%o A306360     for i in range(l):
%o A306360         c = (c + int(s[i])**l) % k
%o A306360     if c == 0:
%o A306360         A306360_list.append(k)
%o A306360     k += 1 # _Chai Wah Wu_, Feb 26 2019
%Y A306360 Cf. A005188, A101337, A306354, A306361.
%K A306360 nonn,base,fini,full
%O A306360 1,2
%A A306360 _Ctibor O. Zizka_, Feb 10 2019
%E A306360 a(22)-a(37) from _Daniel Suteu_, Feb 10 2019