[go: nahoru, domu]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a061383 -id:a061383
Displaying 1-10 of 34 results found.
page 1 2 3 4
     Sort: relevance | references | number | modified | created      Format: long | short | data
A285093
Corresponding values of arithmetic means of digits of numbers from A061383.
+20
1
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Jaroslav Krizek, Table of n, a(n) for n = 0..3000
FORMULA
a(n) = A007953(A061383(n)) / A055642(A061383(n)) for n >= 1.
PROG
(Magma) [&+Intseq(n) / #Intseq(n): n in [1..100000] | &+Intseq(n) mod #Intseq(n) eq 0]
(PARI) 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
CROSSREFS
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).
KEYWORD
nonn,base
AUTHOR
Jaroslav Krizek, Apr 14 2017
STATUS
approved
A004427
Arithmetic mean of digits of n (rounded up).
+10
29
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(100)=1 is the first value that differs from the variant "... rounded to the nearest integer". - M. F. Hasler, May 10 2015
LINKS
R. Zumkeller, Table of n, a(n) for n = 0..25000
Index entries for Colombian or self numbers and related sequences
FORMULA
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)
MATHEMATICA
Ceiling[Mean[IntegerDigits[#]]]&/@Range[0, 110] (* Harvey P. Dale, Aug 29 2014 *)
PROG
(PARI) 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
CROSSREFS
Cf. A178358, A178359, A178361, A178362, ..., A178369, A178401, A178402, A178403, A178404, A178405.
Cf. A000040, A004426, A007953, A055642, A061383, A074462.
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane
STATUS
approved
A061388
Sum of digits = 5 times number of digits.
+10
20
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Delbert L. Johnson, Table of n, a(n) for n = 1..20000
EXAMPLE
186 is a term as the arithmetic mean of the digits is (1+8+6)/3 = 5.
MATHEMATICA
Select[Range[685], Total[x=IntegerDigits[#]]==5*Length[x] &]
PROG
(Magma) [ n: n in [1..700] | &+Intseq(n) eq 5*#Intseq(n) ]; // Bruno Berselli, Jun 30 2011
CROSSREFS
Cf. A061383, A061384, A061385, A061386, A061387, A061423, A061424, A061425.
KEYWORD
nonn,base,easy
AUTHOR
Amarnath Murthy, May 03 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001
STATUS
approved
A061423
Sum of digits = 6 times number of digits.
+10
19
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
288 is a term as the arithmetic mean of the digits is (2+8+8)/3 = 6.
MAPLE
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
MATHEMATICA
Select[Range[1000], Total[IntegerDigits[#]]==6*IntegerLength[#]&] (* Harvey P. Dale, Dec 20 2014 *)
PROG
(PARI) isok(n) = {digs = digits(n, 10); return(6*#digs == sum(k=1, #digs, digs[k])); } \\ Michel Marcus, Jul 31 2013
(Magma) [n: n in [1..1000] | &+Intseq(n) eq 6*#Intseq(n)]; // Vincenzo Librandi, Jan 28 2016
CROSSREFS
Cf. A061383, A061384, A061385, A061386, A061387, A061388, A061424, A061425.
KEYWORD
base,easy,nonn
AUTHOR
Amarnath Murthy, May 03 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001
STATUS
approved
A061425
Sum of digits = 8 times number of digits.
+10
19
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
879 is a term as the arithmetic mean of the digits is (8+7+9)/3 = 8.
MAPLE
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
CROSSREFS
Cf. A061383-A061388, A061423-A061424.
KEYWORD
nonn,base,easy
AUTHOR
Amarnath Murthy, May 03 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001
STATUS
approved
A061384
Numbers n such that sum of digits = number of digits.
+10
16
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
(list; graph; refs; listen; history; text; internal format)
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).
KEYWORD
nonn,base,easy
AUTHOR
Amarnath Murthy, May 03 2001
EXTENSIONS
More terms from Erich Friedman, May 08 2001
STATUS
approved
A004426
Arithmetic mean of digits of n (rounded down).
+10
14
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
From Reinhard Zumkeller, May 27 2010: (Start)
A004427(n) <= a(n);
a(A061383(n)) = A004427(A061383(n));
a(A000040(n)) = A074461(n). (End)
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences
FORMULA
a(n) = floor(A007953(n)/A055642(n)). - Reinhard Zumkeller, May 27 2010
MATHEMATICA
Table[Floor[Mean[IntegerDigits[n]]], {n, 0, 99}] (* Enrique Pérez Herrero, Sep 28 2013 *)
PROG
(Haskell)
a004426 n = (a007953 n) `div` (a055642 n)
-- Reinhard Zumkeller, Jun 18 2013
CROSSREFS
Cf. A175688.
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane
STATUS
approved
A059708
Numbers n such that all digits have same parity.
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A059717(a(n)) = a(n). - Reinhard Zumkeller, Jul 05 2011
A059707(a(n)) = a(n). - Reinhard Zumkeller, Jun 15 2012
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.
MATHEMATICA
Select[Range[0, 140], (cnt = Count[id = IntegerDigits[#], _?OddQ]; cnt == 0 || cnt == Length[id]) & ] (* Jean-François Alcover, May 16 2013 *)
PROG
(Haskell)
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
CROSSREFS
Union of A014261 and A014263.
Cf. A059707, A061383.
KEYWORD
nonn,base,easy,look
AUTHOR
N. J. A. Sloane, Feb 07 2001
STATUS
approved
A061424
Sum of digits = 7 times number of digits.
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
498 is a term as the arithmetic mean of the digits is (4+9+8)/3 = 7.
MAPLE
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
MATHEMATICA
Select[Range[5000], Total[IntegerDigits[#]]==7IntegerLength[#]&] (* Harvey P. Dale, Oct 22 2022 *)
CROSSREFS
Cf. A061383-A061388, A061423-A061425.
KEYWORD
nonn,base,easy
AUTHOR
Amarnath Murthy, May 03 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001
Offset changed by Robert Israel, Dec 07 2020
STATUS
approved
A062179
Harmonic mean of digits is an integer.
+10
8
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
1236 is a term as the harmonic mean is 4/(1+1/2+1/3+1/6) = 2.
MATHEMATICA
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 *)
CROSSREFS
Cf. A062180-A062185, A061383-A061388, A061423-A061425.
KEYWORD
base,easy,nonn
AUTHOR
Vladeta Jovovic, Jun 12 2001
EXTENSIONS
More terms from Robert G. Wilson v, Aug 08 2001
STATUS
approved
page 1 2 3 4

Search completed in 0.016 seconds

Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified November 7 04:03 EST 2024. Contains 377523 sequences.
License Agreements, Terms of Use, Privacy Policy