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A349705 Numbers k such that the concatenation in increasing order of their prime factors, with multiplicity, is congruent to 1 (mod k).

%I A349705 #29 Apr 09 2022 17:38:55
%S A349705 36,39,66,1435,5714,6410,13861,22564,27346,33137,45542,79260,171860,
%T A349705 268218,442068,486127,675423,2287527,3710027,9610766,14318290,
%U A349705 26293568,29361702,49703324,227358366,433100023,442960845,479174118,1221238938,1243718114,4053362596,8620689655
%N A349705 Numbers k such that the concatenation in increasing order of their prime factors, with multiplicity, is congruent to 1 (mod k).
%H A349705 Martin Ehrenstein, <a href="/A349705/b349705.txt">Table of n, a(n) for n = 1..39</a>
%e A349705 a(3) = 66 is a term because the concatenation of its prime factors is 2311 and 2311 == 1 (mod 66).
%p A349705 filter:= proc(n) local L,t;
%p A349705   lcat(map(t -> t[1]$t[2], sort( ifactors(n)[2], (a,b) -> a[1] < b[1]))) mod n = 1;
%p A349705 end proc:
%p A349705 select(filter, [$1..10^7]);
%t A349705 upto=10^5;a={};Do[If[Mod[FromDigits[Flatten[Map[IntegerDigits[ConstantArray[First[#],Last[#]]]&,FactorInteger[k]]]],k]==1,AppendTo[a,k]],{k,upto}];a (* _Paolo Xausa_, Nov 26 2021 *)
%o A349705 (Python)
%o A349705 from sympy import factorint
%o A349705 def ok(k): return int("".join(map(str, factorint(k, multiple=True))))%k == 1
%o A349705 print([k for k in range(2, 10**5) if ok(k)]) # _Michael S. Branicky_, Nov 26 2021
%o A349705 (Python)
%o A349705 from itertools import count, islice
%o A349705 from sympy import factorint
%o A349705 def A349705_gen(startvalue=1): # generator of terms >= startvalue
%o A349705     for k in count(max(startvalue,1)):
%o A349705         c = 0
%o A349705         for d in sorted(factorint(k,multiple=True)):
%o A349705             c = (c*10**len(str(d)) + d) % k
%o A349705         if c == 1:
%o A349705             yield k
%o A349705 A349705_list = list(islice(A349705_gen(),10)) # _Chai Wah Wu_, Feb 28 2022
%Y A349705 Cf. A037276, A259047.
%K A349705 nonn,hard
%O A349705 1,1
%A A349705 _J. M. Bergot_ and _Robert Israel_, Nov 25 2021
%E A349705 a(28)-a(32) from _Martin Ehrenstein_, Nov 27 2021