A384538 Positive integers k >= 10 for which for every pair of nonempty substrings that concatenate to give k one substring divides the other.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 31, 33, 36, 39, 40, 41, 42, 44, 48, 50, 51, 55, 60, 61, 62, 63, 66, 70, 71, 77, 80, 81, 82, 84, 88, 90, 91, 93, 99, 100, 101, 102, 105, 110, 111, 120, 121, 122, 123, 124, 126, 130, 131, 140, 141

Offset: 1

Author

Felix Huber, Jun 09 2025

Keywords

• nonn
• base

Examples

			324 is a term because 3 divides 24 and 4 divides 32.
105 is a term because 1 divides 05 = 5 and 5 divides 10.
2500 is a term because 2 divides 500. 25 divides 00 = 0 and 250 divides 0.
104 is not a term: Although 1 divides 04 = 4, 4 does not divide 10.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..2000

Crossrefs

Supersequence of A384539.

Cf. A102766, A228103.

Programs

Maple

A384538:=proc(n)
    option remember;
    local i,j,k,p,m,q,L;
    if n=1 then
        10
    else
        for k from procname(n-1)+1 do
            L:=ListTools:-Reverse(convert(k,'base',10));
            m:=length(k)-1;
            for j to m do
                p:=parse(cat(seq(L[i],i=1..j)));
                q:=k-p*10^(m+1-length(p));
                if q mod p<>0 and p mod q<>0 then
                    break
            	 elif j=m then
                    return k
                fi
            od
        od
    fi;
end proc;	
seq(A384538(n),n=1..62);

PARI

isok(k) = my(nb=logint(k, 10), d=10); for (i=1, nb, my(sa = k%d, sb=k\d); if ((sa % sb) && (sb % sa), return(0)); d *= 10;); return(1); \\ Michel Marcus, Jun 19 2025

Python

def c(k, m): return (k > 0 and m%k == 0) or (m > 0 and k%m == 0)
def ok(n):
    s = str(n)
    return n > 9 and all(c(int(s[:i]), int(s[i:])) for i in range(1, len(s)))
print([k for k in range(150) if ok(k)]) # Michael S. Branicky, Jun 17 2025