A038447 -id:A038447 - cp's OEIS frontend

A052217 Numbers whose sum of digits is 3.

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

Offset: 1

Henry Bottomley, Feb 01 2000

From Joshua S.M. Weiner, Oct 19 2012: (Start)
Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.
0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)
A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014
Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

Cf. A069521 to A069530, A069532 to A069537.
Cf. A007953, A218043 (subsequence).
Row n=3 of A245062.
Other digit sums: A011557 (1), A052216 (2), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Other bases: A014311 (binary), A226636 (ternary), A179243 (Zeckendorf).
Cf. A242614, A242627.
Cf. A003056, A002262 (triangular coordinates), A056556, A056557, A056558 (tetrahedral coordinates).

a052217 n = a052217_list !! (n-1)
a052217_list = filter ((== 3) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0,3],n],{n,6}],1],Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)
Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

apply( {A052217_row(n,s,t=-1)=vector(n*(n+1)\2,k,t++>s&&t=!s++;10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))
print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A052217(n): return 10**((m:=integer_nthroot(6*n,3)[0])-(a:=n<=comb(m+2,3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)),2)) # Chai Wah Wu, Dec 11 2024

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

Offset changed from 0 to 1 by Vincenzo Librandi, Mar 07 2013

A014313 Numbers with exactly 5 ones in binary expansion.

Original entry on oeis.org

31, 47, 55, 59, 61, 62, 79, 87, 91, 93, 94, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217, 218, 220, 227, 229, 230, 233, 234, 236, 241, 242

Offset: 1

Al Black (gblack(AT)nol.net)

Appears to give all n such that 4096 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002

T. D. Noe, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.
Index entries for 2-automatic sequences.

Cf. A000079, A018900, A014311, A014312, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9).

Cf. A005148, A007088, A038447, A057168.

a014313 = f . a038447 where
   f x = if x == 0 then 0 else 2 * f x' + b  where (x', b) = divMod x 10
-- Reinhard Zumkeller, Jan 06 2015

Select[ Range[31, 240], Total[IntegerDigits[#, 2]] == 5&]

sum_of_bits(n) = if(n<1, 0, sum_of_bits(floor(n/2))+n%2)
isA014313(n) = (sum_of_bits(n) == 5); \\ Michael B. Porter, Oct 21 2009

is(n)=hammingweight(n)==5 \\ Charles R Greathouse IV, Nov 17 2013

print1(t=2^5-1); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 2014

from itertools import islice
def A014313_gen(): # generator of terms
    yield (n:=31)
    while True: yield (n:=((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014313_list = list(islice(A014313_gen(),30)) # Chai Wah Wu, Mar 06 2025

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014
A038447(n) = A007088(a(n)). - Reinhard Zumkeller, Jan 06 2015
Sum_{n>=1} 1/a(n) = 1.390704528210321982529622080740025763242354253694629591331835888395977392151... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extension and program by Olivier Gérard

A038445 Sums of 3 distinct powers of 10.

Original entry on oeis.org

111, 1011, 1101, 1110, 10011, 10101, 10110, 11001, 11010, 11100, 100011, 100101, 100110, 101001, 101010, 101100, 110001, 110010, 110100, 111000, 1000011, 1000101, 1000110, 1001001, 1001010, 1001100, 1010001, 1010010, 1010100, 1011000, 1100001, 1100010, 1100100

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557.

Cf. A038444, A038446, A038447, A038448, A038449, A038450, A038451, A038452, A038453, A038454.

Sort[Plus @@@ Subsets[10^Range[0, 6], {3}]] (* Amiram Eldar, Jul 12 2022 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A038445(n): return 10**((r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+10**((a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+10**(m+t+1) # Chai Wah Wu, Mar 10 2025

A038446 Sums of 4 distinct powers of 10.

Original entry on oeis.org

1111, 10111, 11011, 11101, 11110, 100111, 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100, 1000111, 1001011, 1001101, 1001110, 1010011, 1010101, 1010110, 1011001, 1011010, 1011100, 1100011, 1100101, 1100110, 1101001, 1101010, 1101100, 1110001

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557, A157711 (primes subsequence).

Cf. A038444, A038445, A038447, A038448, A038449, A038450, A038451, A038452, A038453, A038454.

Total/@Subsets[10^Range[0,6],{4}]//Union (* Harvey P. Dale, Nov 07 2021 *)

from itertools import islice
def A038446_gen(): # generator of terms
    yield int(bin(n:=15)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038446_list = list(islice(A038446_gen(),20)) # Chai Wah Wu, Mar 11 2025

Offset corrected by Amiram Eldar, Jul 12 2022

A038452 Sums of 10 distinct powers of 10.

Original entry on oeis.org

1111111111, 10111111111, 11011111111, 11101111111, 11110111111, 11111011111, 11111101111, 11111110111, 11111111011, 11111111101, 11111111110, 100111111111, 101011111111, 101101111111, 101110111111, 101111011111, 101111101111, 101111110111, 101111111011, 101111111101

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557.

Cf. A038444, A038445, A038446, A038447, A038448, A038449, A038450, A038451, A038453, A038454.

Take[Union[Total/@Subsets[10^Range[0,15],{10}]],20] (* Harvey P. Dale, Dec 19 2011 *)

from itertools import islice
def A038452_gen(): # generator of terms
    yield int(bin(n:=1023)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038452_list = list(islice(A038452_gen(),20)) # Chai Wah Wu, Mar 10 2025

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A038448 Sums of 6 distinct powers of 10.

Original entry on oeis.org

111111, 1011111, 1101111, 1110111, 1111011, 1111101, 1111110, 10011111, 10101111, 10110111, 10111011, 10111101, 10111110, 11001111, 11010111, 11011011, 11011101, 11011110, 11100111, 11101011, 11101101, 11101110, 11110011, 11110101, 11110110, 11111001, 11111010

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557.

Cf. A038444, A038445, A038446, A038447, A038449, A038450, A038451, A038452, A038453, A038454.

Sort[Plus @@@ Subsets[10^Range[0, 7], {6}]] (* Amiram Eldar, Jul 12 2022 *)

from itertools import islice
def A038448_gen(): # generator of terms
    yield int(bin(n:=63)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038448_list = list(islice(A038448_gen(),20)) # Chai Wah Wu, Mar 11 2025

Offset corrected by Amiram Eldar, Jul 12 2022

A038449 Sums of 7 distinct powers of 10.

Original entry on oeis.org

1111111, 10111111, 11011111, 11101111, 11110111, 11111011, 11111101, 11111110, 100111111, 101011111, 101101111, 101110111, 101111011, 101111101, 101111110, 110011111, 110101111, 110110111, 110111011, 110111101, 110111110, 111001111, 111010111, 111011011, 111011101

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557.

Cf. A038444, A038445, A038446, A038447, A038448, A038450, A038451, A038452, A038453, A038454.

Take[Total/@Subsets[10^Range[0,20],{7}]//Union,20] (* Harvey P. Dale, Feb 25 2018 *)

from itertools import islice
def A038449_gen(): # generator of terms
    yield int(bin(n:=127)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038449_list = list(islice(A038449_gen(),20)) # Chai Wah Wu, Mar 11 2025

Offset corrected by Amiram Eldar, Jul 12 2022

A038453 Sums of 11 distinct powers of 10.

Original entry on oeis.org

11111111111, 101111111111, 110111111111, 111011111111, 111101111111, 111110111111, 111111011111, 111111101111, 111111110111, 111111111011, 111111111101, 111111111110, 1001111111111, 1010111111111, 1011011111111, 1011101111111, 1011110111111, 1011111011111, 1011111101111

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557.

Cf. A038444, A038445, A038446, A038447, A038448, A038449, A038450, A038451, A038452, A038454.

Union[Total/@Subsets[10^Range[0,12],{11}]] (* Harvey P. Dale, Jan 20 2013 *)

lista(nn) = {for (n=1, nn, if (hammingweight(n) == 11, print1(subst(Pol(binary(n)), x, 10), ", ");););} \\ Michel Marcus, Feb 29 2016

from itertools import islice
def A038453_gen(): # generator of terms
    yield int(bin(n:=2047)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038453_list = list(islice(A038453_gen(),20)) # Chai Wah Wu, Mar 11 2025

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A038454 Sums of 12 distinct powers of 10.

Original entry on oeis.org

111111111111, 1011111111111, 1101111111111, 1110111111111, 1111011111111, 1111101111111, 1111110111111, 1111111011111, 1111111101111, 1111111110111, 1111111111011, 1111111111101, 1111111111110, 10011111111111, 10101111111111, 10110111111111, 10111011111111, 10111101111111

Offset: 1

Olivier Gérard

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

Cf. A011557.

Cf. A038444, A038445, A038446, A038447, A038448, A038449, A038450, A038451, A038452, A038453.

N:= 14: # to get all terms of at most N digits
sort(map(t -> (10^N-1)/9 - add(10^j, j=t),
combinat:-choose([$0..N-1],N-12))); # Robert Israel, Feb 28 2016

Sort[Plus @@@ Subsets[10^Range[0, 12], {12}]] (* Amiram Eldar, Jul 12 2022 *)

from itertools import islice
def A038454_gen(): # generator of terms
    yield int(bin(n:=4095)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038454_list = list(islice(A038454_gen(),20)) # Chai Wah Wu, Mar 11 2025

a(binomial(N,12)+k) = 10^N + A038453(k) for 1 <= k <= binomial(N,11). - Robert Israel, Feb 28 2016

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A038450 Sums of 8 distinct powers of 10.

Original entry on oeis.org

11111111, 101111111, 110111111, 111011111, 111101111, 111110111, 111111011, 111111101, 111111110, 1001111111, 1010111111, 1011011111, 1011101111, 1011110111, 1011111011, 1011111101, 1011111110, 1100111111, 1101011111, 1101101111, 1101110111, 1101111011, 1101111101

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011557.

Cf. A038444, A038445, A038446, A038447, A038448, A038449, A038451, A038452, A038453, A038454.

Sort[FromDigits/@Permutations[{1,1,1,1,1,1,1,1,0,0}]] (* Harvey P. Dale, Apr 29 2013 *)

from itertools import islice
def A038450_gen(): # generator of terms
    yield int(bin(n:=255)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038450_list = list(islice(A038450_gen(),20)) # Chai Wah Wu, Mar 11 2025

Offset corrected by Amiram Eldar, Jul 12 2022

cp's OEIS Frontend

A052217 Numbers whose sum of digits is 3.

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A014313 Numbers with exactly 5 ones in binary expansion.

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A038445 Sums of 3 distinct powers of 10.

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A038446 Sums of 4 distinct powers of 10.

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A038452 Sums of 10 distinct powers of 10.

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A038448 Sums of 6 distinct powers of 10.

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A038449 Sums of 7 distinct powers of 10.

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