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

1,2

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

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A071976.

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

Sequence in context: A180113 A198310 A085187 * A071154 A071161 A367609

Adjacent sequences: A061381 A061382 A061383 * A061385 A061386 A061387

Keyword

nonn,base,easy

Author

Amarnath Murthy, May 03 2001

Extensions

More terms from Erich Friedman, May 08 2001

Status

approved