A061384 - OEIS

%I #59 Nov 28 2023 22:00:13

%S 1,11,20,102,111,120,201,210,300,1003,1012,1021,1030,1102,1111,1120,

%T 1201,1210,1300,2002,2011,2020,2101,2110,2200,3001,3010,3100,4000,

%U 10004,10013,10022,10031,10040,10103,10112,10121,10130,10202,10211

%N Numbers n such that sum of digits = number of digits.

%C Number of d-digit entries is A071976(d). - _Robert Israel_, Apr 06 2016

%C Equivalently, numbers n > 0 for which the arithmetic mean of the digits equals 1. - _M. F. Hasler_, Dec 07 2018

%H Robert Israel, <a href="/A061384/b061384.txt">Table of n, a(n) for n = 1..10000</a>

%F {n > 0 | A007953(n) = A055642(n)}. - _M. F. Hasler_, Dec 07 2018

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

%p Q:= proc(n,s) option remember;

%p # n-digit integers with digit sum s

%p if s = 0 then []

%p elif s = 1 then [10^(n-1)]

%p elif n = 1 then

%p    if s <= 9 then [s]

%p    else []

%p    fi

%p else

%p   map(op,[seq(map(t -> 10*t+i, procname(n-1,s-i)), i=0..min(9,s-1))])

%p fi

%p end proc:

%p map(op, [seq(sort(Q(n,n)),n=1..5)]); # _Robert Israel_, Apr 06 2016

%t Select[Range[15000], Total[IntegerDigits[#]] == IntegerLength[#]&] (* _Harvey P. Dale_, Jan 08 2011 *)

%o (Magma) [ n: n in [1..10215] | &+Intseq(n) eq #Intseq(n) ]; // _Bruno Berselli_, Jun 30 2011

%o (PARI) isok(n) = (sumdigits(n)/#Str(n) == 1); \\ _Michel Marcus_, Mar 28 2016

%o (PARI) is_A061384(n)={sumdigits(n)==logint(n+!n,10)+1} \\ _M. F. Hasler_, Dec 07 2018

%o (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

%o (Python)

%o from itertools import count, islice

%o def Q(n, s): # length-n strings of 0..9 with sum s, after _Robert Israel_

%o     if s == 0: yield "0"*n

%o     elif n == 1: yield (str(s) if s <= 9 else "")

%o     else:

%o         m = min(9, s) + 1

%o         yield from (str(i)+t for i in range(m) for t in Q(n-1, s-i))

%o def agen():

%o     yield from (int(t) for n in count(1) for t in Q(n, n) if t[0] != "0")

%o print(list(islice(agen(), 43))) # _Michael S. Branicky_, May 26 2022

%o (Python)

%o from itertools import count, islice

%o from collections import Counter

%o from sympy.utilities.iterables import partitions, multiset_permutations

%o def A061384_gen(): # generator of terms

%o     for l in count(1):

%o         for i in range(1,min(l,9)+1):

%o             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())))

%o A061384_list = list(islice(A061384_gen(),30)) # _Chai Wah Wu_, Nov 28 2023

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

%Y Cf. A071976.

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

%K nonn,base,easy

%O 1,2

%A _Amarnath Murthy_, May 03 2001

%E More terms from _Erich Friedman_, May 08 2001