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A348306 List of Agathokakological Numbers "k": string of digits of the juxtaposition of the prime factors of k has the same length as k but these digits do not appear in k.

%I A348306 #71 May 10 2022 02:11:11
%S A348306 10,14,21,49,106,111,118,129,134,146,158,161,166,177,201,219,249,259,
%T A348306 267,329,343,413,511,553,623,1011,1029,1046,1077,1081,1101,1106,1114,
%U A348306 1119,1138,1149,1167,1186,1227,1299,1318,1354,1358,1363,1418,1454,1466,1538,1541,1546,1561,1589,1591
%N A348306 List of Agathokakological Numbers "k": string of digits of the juxtaposition of the prime factors of k has the same length as k but these digits do not appear in k.
%C A348306 Theorem: (See PDF "PROOFS" in Links)
%C A348306 Of Agathokakological Numbers k,
%C A348306 No k have a leading 9.
%C A348306 No k end in 2 or 5.
%C A348306 10 is the only k to end in 0. It is also the only k with 5 as a prime factor.
%C A348306 Can only be square terms when k is of the order 10^m where m is odd.
%C A348306 For k written as a*10^m, k can only be even when 1<=a<1.888...
%C A348306 Empirical observation: When graphed with the log of the n-th term on x axis and the log of the n-th term's value on the y axis a pattern appears with a similar shape for each new power of ten (see figure "LogLogGraph" in Links)
%C A348306 Special cases 28651 = 7*4093 and 65821 = 7*9043 use all digits 0-9 once.
%C A348306 "Agathokakological" is a Greek word meaning "composed of both good and evil." (Merriam-Webster) The composition (prime factorization) of Agathokakological Numbers is both good (same length) and evil (no common digits).
%H A348306 Samuel Harkness, <a href="/A348306/b348306.txt">Table of n, a(n) for n = 1..6388</a>
%H A348306 Samuel Harkness, <a href="/A348306/a348306.m.txt">MATLAB</a>
%H A348306 Samuel Harkness, <a href="/A348306/a348306.jpg">LogLogGraph</a>
%H A348306 Samuel Harkness, <a href="/A348306/a348306.pdf">PROOFS</a>
%e A348306 158 = 2 * 79 since {2,7,9} do not appear in {1,5,8} and both have 3 digits.
%t A348306 q[n_] := Module[{d = IntegerDigits[n], f = FactorInteger[n]}, Length[d] == Plus @@ ((Last[#]*IntegerLength[First[#]]) & /@ f ) && Intersection[d, Join @@ IntegerDigits[f[[;; , 1]]]] == {}]; Select[Range[1600], q] (* _Amiram Eldar_, Oct 12 2021 *)
%o A348306 (PARI) digsf(n) = my(f=factor(n), list=List()); for (k=1, #f~, my(dk=digits(f[k,1])); for (i=1, f[k,2], for (j=1, #dk, listput(list, dk[j])))); Vec(list);
%o A348306 isokd(m) = my(df=digsf(m), d=digits(m)); (#df == #d) && (#setintersect(Set(df), Set(d)) == 0); \\ _Michel Marcus_, Oct 11 2021
%o A348306 (Python)
%o A348306 from sympy import factorint
%o A348306 def ok(n):
%o A348306     s, f = str(n), factorint(n)
%o A348306     pfd = set("".join(str(p) for p in f))
%o A348306     if set(s) & pfd != set(): return False
%o A348306     return len(s) == sum(len(str(p))*f[p] for p in f)
%o A348306 print(list(filter(ok, range(1601)))) # _Michael S. Branicky_, Oct 11 2021
%Y A348306 Cf. A055642, A076649, A280928.
%Y A348306 Intersection of A035139 and A109608.
%Y A348306 Subsequence of A047201 from n=2.
%K A348306 nonn,base
%O A348306 1,1
%A A348306 _Samuel Harkness_, Oct 11 2021