A359959 a(n) is the least number that has exactly n divisors with the same digit sum.

1, 10, 36, 54, 144, 108, 486, 216, 324, 432, 648, 540, 1296, 3510, 2430, 1080, 2700, 1620, 8424, 2160, 4860, 4320, 3240, 27216, 7560, 8100, 6480, 35100, 10800, 19440, 24300, 21060, 15120, 16200, 37800, 56700, 54000, 30240, 42120, 60480, 32400, 45360, 84240, 81000, 64800, 75600, 90720, 213840

Offset

1,2

Comments

a(n) is the least number k such that for some s, there are exactly n divisors of k with sum of digits s.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..289

Michael S. Branicky, Table of n, a(n) for all terms <= 446000000

Formula

a(n) <= 10^(n-1). - Rémy Sigrist, Jan 27 2023

Example

a(1) = 1 because 1 has 1 divisor with digit sum 1, namely 1.
a(2) = 10 because 10 has 2 divisors with digit sum 1, namely 1 and 10.
a(3) = 36 because 36 has 3 divisors with digit sum 9, namely 9, 18 and 36.
a(4) = 54 because 54 has 4 divisors with digit sum 9, namely 9, 18, 27 and 54.

Maple

f:= proc(n) local L, S;
      L:= convert(numtheory:-divisors(n), list);
      S:= map(t -> convert(convert(t, base, 10), `+`), L);
      map(t -> numboccur(t, S), convert(S, set))
end proc:
V:= Vector(50): count:= 0:
for n from 1 while count < 50 do
   for v in f(n) do
     if v <= 50 and V[v] = 0 then V[v]:= n; count:= count+1; fi
od od:
convert(V, list);

Prog

(PARI) is(z, n)={my(e=1, w=[], s=[], t=0); s=vecsort(apply(vecsum, apply(digits, apply(divisors, z)))); for(i=2, #s, if(s[i]==s[i-1], e++, w=concat(w, e); e=1)); w=concat(w, e); s=Set(w); forvec(G=vector(2, j, [1, #s]), if((s[G[1]]==n)&&(s[G[2]]==n), t=1; if(G[1]!=G[2], return(0)))); return(t)}
a(n)=for(z=1, +oo, is(z, n)&&return(z)); \\ R. J. Cano, Jan 23 2023
(Python)
from sympy import divisors
from collections import Counter
from itertools import count, islice
def sd(n): return sum(map(int, str(n)))
def agen(): # generator of terms
    adict, n = dict(), 1
    for k in count(1):
        c = Counter(sd(d) for d in divisors(k, generator=True))
        for v in c.values():
            if v >= n and v not in adict:
                adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Jan 27 2023

Crossrefs

Cf. A007953, A359074.

Sequence in context: A073613 A346386 A117404 * A309783 A072517 A271912

Adjacent sequences: A359956 A359957 A359958 * A359960 A359961 A359962

Keyword

nonn,base

Author

Robert Israel, Jan 19 2023

Status

approved