A061384 Numbers n such that sum of digits = number of digits.

1, 11, 20, 102, 111, 120, 201, 210, 300, 1003, 1012, 1021, 1030, 1102, 1111, 1120, 1201, 1210, 1300, 2002, 2011, 2020, 2101, 2110, 2200, 3001, 3010, 3100, 4000, 10004, 10013, 10022, 10031, 10040, 10103, 10112, 10121, 10130, 10202, 10211

Offset: 1

Author

Amarnath Murthy, May 03 2001

Keywords

• nonn
• base
• easy

Comments

Number of d-digit entries is A071976(d). - Robert Israel, Apr 06 2016

Equivalently, numbers n > 0 for which the arithmetic mean of the digits equals 1. - M. F. Hasler, Dec 07 2018

Examples

			120 is a term as the arithmetic mean of the digits is (1+2+0)/3 = 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Totally balanced subset: A071154. Cf. also A061383-A061388, A061423-A061425.

Cf. A071976.

Cf. A007953 (sum of digits), A055642 (number of digits).

Programs

Magma

[ n: n in [1..10215] | &+Intseq(n) eq #Intseq(n) ]; // Bruno Berselli, Jun 30 2011

Maple

Q:= proc(n,s) option remember;
# n-digit integers with digit sum s
if s = 0 then []
elif s = 1 then [10^(n-1)]
elif n = 1 then
   if s <= 9 then [s]
   else []
   fi
else
  map(op,[seq(map(t -> 10*t+i, procname(n-1,s-i)), i=0..min(9,s-1))])
fi
end proc:
map(op, [seq(sort(Q(n,n)),n=1..5)]); # Robert Israel, Apr 06 2016

Mathematica

Select[Range[15000], Total[IntegerDigits[#]] == IntegerLength[#]&] (* Harvey P. Dale, Jan 08 2011 *)

PARI

isok(n) = (sumdigits(n)/#Str(n) == 1); \\ Michel Marcus, Mar 28 2016

PARI

is_A061384(n)={sumdigits(n)==logint(n+!n,10)+1} \\ M. F. Hasler, Dec 07 2018

PARI

A061384_row(n)={my(L=List(), u=vector(n, i, i==1), d); forvec(v=vector(n+1, i, [if(i>n,n, 1), if(i>1, n, 1)]), vecmax(d=v[^1]-v[^-1]+u)<10 && listput(L,fromdigits(d)),1);Vec(L)} \\ Return the list of all n-digit terms. - M. F. Hasler, Dec 07 2018

Python

from itertools import count, islice
def Q(n, s): # length-n strings of 0..9 with sum s, after Robert Israel
    if s == 0: yield "0"*n
    elif n == 1: yield (str(s) if s <= 9 else "")
    else:
        m = min(9, s) + 1
        yield from (str(i)+t for i in range(m) for t in Q(n-1, s-i))
def agen():
    yield from (int(t) for n in count(1) for t in Q(n, n) if t[0] != "0")
print(list(islice(agen(), 43))) # Michael S. Branicky, May 26 2022

Python

from itertools import count, islice
from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
def A061384_gen(): # generator of terms
    for l in count(1):
        for i in range(1,min(l,9)+1):
            yield from sorted(int(str(i)+''.join(map(str,j))) for s,p in partitions(l-i,k=9,size=True) for j in multiset_permutations([0]*(l-1-s)+list(Counter(p).elements())))
A061384_list = list(islice(A061384_gen(),30)) # Chai Wah Wu, Nov 28 2023

Formula

{n > 0 | A007953(n) = A055642(n)}. - M. F. Hasler, Dec 07 2018

Extensions

More terms from Erich Friedman, May 08 2001