A061383 -id:A061383 - cp's OEIS frontend

A285093 Corresponding values of arithmetic means of digits of numbers from A061383.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6

Offset: 0

Jaroslav Krizek, Apr 14 2017

Jaroslav Krizek, Table of n, a(n) for n = 0..3000

Cf. A061383 (numbers with integer arithmetic mean of digits in base 10).
Sequences of numbers n such that a(n) = k for k = 1 - 9: A061384 (k = 1), A061385 (k = 2), A061386 (k = 3), A061387 (k = 4), A061388 (k = 5), A061423 (k = 6), A061424 (k = 7), A061425 (k = 8), A002283 (k = 9).
Cf. A004426, A004427, A257295 (supersequences).

[0] cat [&+Intseq(n) / #Intseq(n): n in [1..100000] | &+Intseq(n) mod #Intseq(n) eq 0];

lista(nn) = {for (n=0, nn, if (n, d = digits(n), d = [0]); if (!( vecsum(d) % #d), print1(vecsum(d)/#d, ", ")););} \\ Michel Marcus, Apr 15 2017

a(n) = A007953(A061383(n)) / A055642(A061383(n)) for n >= 1.

A004427 Arithmetic mean of digits of n (rounded up).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 1, 1, 1, 2, 2, 2, 3, 3

Offset: 0

N. J. A. Sloane

a(100)=1 is the first value that differs from the variant "... rounded to the nearest integer". - M. F. Hasler, May 10 2015

Cf. A178358, A178359, A178361, A178362, ..., A178369, A178401, A178402, A178403, A178404, A178405.

Cf. A000040, A004426, A007953, A055642, A061383, A074462.

Ceiling[Mean[IntegerDigits[#]]]&/@Range[0,110] (* Harvey P. Dale, Aug 29 2014 *)

A004427(n)=ceil(sum(i=1, #n=digits(n), n[i])/#n) \\ ...Vecsmall(Str(n))...-48 is a little faster. \\ M. F. Hasler, May 10 2015

From Reinhard Zumkeller, May 27 2010: (Start)
a(n) = ceiling(A007953(n)/A055642(n)); a(A000040(n)) = A074462(n);
A004426(n) <= a(n) with equality for n in A061383;
a(A178361(n)) = 1; a(A178362(n)) = 2; a(A178363(n)) = 3; a(A178364(n)) = 4; a(A178365(n)) = 5; a(A178366(n)) = 6; a(A178367(n)) = 7; a(A178368(n)) = 8; a(A178369(n)) = 9. (End)

A061388 Sum of digits = 5 times number of digits.

Original entry on oeis.org

5, 19, 28, 37, 46, 55, 64, 73, 82, 91, 159, 168, 177, 186, 195, 249, 258, 267, 276, 285, 294, 339, 348, 357, 366, 375, 384, 393, 429, 438, 447, 456, 465, 474, 483, 492, 519, 528, 537, 546, 555, 564, 573, 582, 591, 609, 618, 627, 636, 645, 654, 663, 672, 681

Offset: 1

Amarnath Murthy, May 03 2001

			186 is a term as the arithmetic mean of the digits is (1+8+6)/3 = 5.

Delbert L. Johnson, Table of n, a(n) for n = 1..20000

Cf. A061383, A061384, A061385, A061386, A061387, A061423, A061424, A061425.

[ n: n in [1..700] | &+Intseq(n) eq 5*#Intseq(n) ];  // Bruno Berselli, Jun 30 2011

Select[Range[685],Total[x=IntegerDigits[#]]==5*Length[x] &]

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

A061423 Sum of digits = 6 times number of digits.

Original entry on oeis.org

6, 39, 48, 57, 66, 75, 84, 93, 189, 198, 279, 288, 297, 369, 378, 387, 396, 459, 468, 477, 486, 495, 549, 558, 567, 576, 585, 594, 639, 648, 657, 666, 675, 684, 693, 729, 738, 747, 756, 765, 774, 783, 792, 819, 828, 837, 846, 855, 864, 873, 882, 891, 909

Offset: 1

Amarnath Murthy, May 03 2001

			288 is a term as the arithmetic mean of the digits is (2+8+8)/3 = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A061383, A061384, A061385, A061386, A061387, A061388, A061424, A061425.

[n: n in [1..1000] | &+Intseq(n) eq 6*#Intseq(n)]; // Vincenzo Librandi, Jan 28 2016

F:= proc(m,s)
option remember;
# list of all m-digit numbers with sum of digits s
if s > 9*m or s < 0 then return [] fi;
if m = 1 then return [s] fi;
[seq(seq(op(map(`+`,procname(j,s-i),10^(m-1)*i)),j=1..m-1),i=1..min(9,s))]
end proc:
seq(op(F(m,6*m)),m=1..3); # Robert Israel, Jan 27 2016

Select[Range[1000],Total[IntegerDigits[#]]==6*IntegerLength[#]&] (* Harvey P. Dale, Dec 20 2014 *)

isok(n) = {digs = digits(n, 10); return(6*#digs == sum(k=1, #digs, digs[k]));} \\ Michel Marcus, Jul 31 2013

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

A061425 Sum of digits = 8 times number of digits.

Original entry on oeis.org

8, 79, 88, 97, 699, 789, 798, 879, 888, 897, 969, 978, 987, 996, 5999, 6899, 6989, 6998, 7799, 7889, 7898, 7979, 7988, 7997, 8699, 8789, 8798, 8879, 8888, 8897, 8969, 8978, 8987, 8996, 9599, 9689, 9698, 9779, 9788, 9797, 9869, 9878, 9887, 9896, 9959

Offset: 1

Amarnath Murthy, May 03 2001

			879 is a term as the arithmetic mean of the digits is (8+7+9)/3 = 8.

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

Cf. A061383-A061388, A061423-A061424.

Q:= proc(n,t) option remember; local j,R;
      if t > 9*n or t <= 0 then return [] fi;
      R:= NULL;
      for j from 0 to min(t,9) do
        R:= R, op(map(s -> 10*s+j, procname(n-1,t-j)))
      od;
      [R]
end proc;
for i from 0 to 9 do Q(1,i):= [i] od:
seq(op(sort(Q(d,8*d))),d=1..4); # Robert Israel, Dec 07 2020

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

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

Original entry on oeis.org

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

Amarnath Murthy, May 03 2001

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

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

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

Totally balanced subset: A071154. Cf. also A061383-A061388, A061423-A061425.
Cf. A071976.
Cf. A007953 (sum of digits), A055642 (number of digits).

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

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

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

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

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

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

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

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

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

More terms from Erich Friedman, May 08 2001

A059708 Numbers k such that all digits have same parity.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 111, 113, 115, 117, 119, 131, 133, 135, 137, 139

Offset: 1

N. J. A. Sloane, Feb 07 2001

A059717(a(n)) = a(n). - Reinhard Zumkeller, Jul 05 2011

A059707(a(n)) = a(n). - Reinhard Zumkeller, Jun 15 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Union of A014261 and A014263.

Cf. A059707, A059717, A061383.

a059708 n = a059708_list !! (n-1)
a059708_list = filter sameParity [0..] where
   sameParity n = all (`elem` "02468") ns
               || all (`elem` "13579") ns where ns = show n
-- Reinhard Zumkeller, Jul 05 2011

Select[Range[0, 140], (cnt = Count[id = IntegerDigits[#], ?OddQ]; cnt == 0 || cnt == Length[id]) & ] (* _Jean-François Alcover, May 16 2013 *)

A004426 Arithmetic mean of digits of n (rounded down).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 0, 0, 1, 1, 1, 2, 2, 2

Offset: 0

N. J. A. Sloane

From Reinhard Zumkeller, May 27 2010: (Start)
A004427(n) <= a(n);
a(A061383(n)) = A004427(A061383(n));
a(A000040(n)) = A074461(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A175688.

a004426 n = (a007953 n) `div` (a055642 n)
-- Reinhard Zumkeller, Jun 18 2013

Table[Floor[Mean[IntegerDigits[n]]], {n, 0, 99}] (* Enrique Pérez Herrero, Sep 28 2013 *)

a(n) = floor(A007953(n)/A055642(n)). - Reinhard Zumkeller, May 27 2010

A061424 Sum of digits = 7 times number of digits.

Original entry on oeis.org

7, 59, 68, 77, 86, 95, 399, 489, 498, 579, 588, 597, 669, 678, 687, 696, 759, 768, 777, 786, 795, 849, 858, 867, 876, 885, 894, 939, 948, 957, 966, 975, 984, 993, 1999, 2899, 2989, 2998, 3799, 3889, 3898, 3979, 3988, 3997, 4699, 4789, 4798, 4879, 4888, 4897

Offset: 1

Amarnath Murthy, May 03 2001

			498 is a term as the arithmetic mean of the digits is (4+9+8)/3 = 7.

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

Cf. A061383-A061388, A061423-A061425.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  convert(L,`+`)=nops(L)*7
end proc:
select(filter, [$1..10000]); # Robert Israel, Dec 07 2020

Select[Range[5000],Total[IntegerDigits[#]]==7IntegerLength[#]&] (* Harvey P. Dale, Oct 22 2022 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

Offset changed by Robert Israel, Dec 07 2020

A062179 Harmonic mean of digits is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 26, 33, 36, 44, 55, 62, 63, 66, 77, 88, 99, 111, 136, 144, 163, 222, 236, 244, 263, 288, 316, 326, 333, 346, 361, 362, 364, 414, 424, 436, 441, 442, 444, 463, 488, 555, 613, 623, 631, 632, 634, 643, 666, 777, 828, 848, 882, 884

Offset: 1

Vladeta Jovovic, Jun 12 2001

			1236 is a term as the harmonic mean is 4/(1+1/2+1/3+1/6) = 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A062180-A062185, A061383-A061388, A061423-A061425.

Do[ h = IntegerDigits[n]; If[ Sort[h] [[1]] != 0 && IntegerQ[ Length[h] / Apply[ Plus, 1/h] ], Print[n]], {n, 1, 10^4} ] Note that the number of entries <= 10^n are 9, 22, 61, 198, 927, 4738, 24620, 130093,
hmdiQ[n_]:=DigitCount[n,10,0]==0&&IntegerQ[HarmonicMean[ IntegerDigits[ n]]]; Select[Range[1000],hmdiQ] (* Harvey P. Dale, Sep 22 2012 *)

More terms from Robert G. Wilson v, Aug 08 2001

cp's OEIS Frontend

A285093 Corresponding values of arithmetic means of digits of numbers from A061383.

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A004427 Arithmetic mean of digits of n (rounded up).

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A061388 Sum of digits = 5 times number of digits.

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A061423 Sum of digits = 6 times number of digits.

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A061425 Sum of digits = 8 times number of digits.

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

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A059708 Numbers k such that all digits have same parity.

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