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.

10, 14, 21, 49, 106, 111, 118, 129, 134, 146, 158, 161, 166, 177, 201, 219, 249, 259, 267, 329, 343, 413, 511, 553, 623, 1011, 1029, 1046, 1077, 1081, 1101, 1106, 1114, 1119, 1138, 1149, 1167, 1186, 1227, 1299, 1318, 1354, 1358, 1363, 1418, 1454, 1466, 1538, 1541, 1546, 1561, 1589, 1591

Offset: 1

Author

Samuel Harkness, Oct 11 2021

Keywords

• nonn
• base

Comments

Theorem: (See PDF "PROOFS" in Links)

Of Agathokakological Numbers k,

No k have a leading 9.

No k end in 2 or 5.

10 is the only k to end in 0. It is also the only k with 5 as a prime factor.

Can only be square terms when k is of the order 10^m where m is odd.

For k written as a*10^m, k can only be even when 1<=a<1.888...

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)

Special cases 28651 = 7*4093 and 65821 = 7*9043 use all digits 0-9 once.

"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).

Examples

			158 = 2 * 79 since {2,7,9} do not appear in {1,5,8} and both have 3 digits.
		

Links

• Samuel Harkness, Table of n, a(n) for n = 1..6388
• Samuel Harkness, MATLAB
• Samuel Harkness, LogLogGraph
• Samuel Harkness, PROOFS

Crossrefs

Cf. A055642, A076649, A280928.

Intersection of A035139 and A109608.

Subsequence of A047201 from n=2.

Programs

Mathematica

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 *)

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);
isokd(m) = my(df=digsf(m), d=digits(m)); (#df == #d) && (#setintersect(Set(df), Set(d)) == 0); \\ Michel Marcus, Oct 11 2021

Python

from sympy import factorint
def ok(n):
    s, f = str(n), factorint(n)
    pfd = set("".join(str(p) for p in f))
    if set(s) & pfd != set(): return False
    return len(s) == sum(len(str(p))*f[p] for p in f)
print(list(filter(ok, range(1601)))) # Michael S. Branicky, Oct 11 2021