A038444 -id:A038444 - cp's OEIS frontend

A052216 Sums of two powers of 10.

2, 11, 20, 101, 110, 200, 1001, 1010, 1100, 2000, 10001, 10010, 10100, 11000, 20000, 100001, 100010, 100100, 101000, 110000, 200000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000

Offset: 1

Henry Bottomley, Feb 01 2000

Numbers whose digit sum is 2.
A007953(a(n)) = 2; number of repdigits = #{2,11} = A242627(2) = 2. - Reinhard Zumkeller, Jul 17 2014
By extension, numbers k such that digitsum(k)^2 - 1 is prime. (PROOF: For any number k whose digit sum d > 2, d^2 - 1 = (d+1)*(d-1) and thus is not prime.) - Christian N. K. Anderson, Apr 22 2024

			From _Bruno Berselli_, Mar 07 2013: (Start)
The triangular array starts (see formula):
        2;
       11,      20;
      101,     110,     200;
     1001,    1010,    1100,    2000;
    10001,   10010,   10100,   11000,   20000;
   100001,  100010,  100100,  101000,  110000,  200000;
  1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000;
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms 1..48 from Vincenzo Librandi, terms 49..1036 from T. D. Noe)

Subsequence of A069263 and A107679. A038444 is a subsequence.
Sums of n powers of 10: A011557 (1), A052217 (3), 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).
Cf. A002262, A003056, A007953, A242614, A242627, A237424.

a052216 n = a052216_list !! (n-1)
a052216_list = 2 : f [2] 9 where
   f xs@(x:_) z = ys ++ f ys (10 * z) where
                  ys = (x + z) : map (* 10) xs
-- Reinhard Zumkeller, Jan 28 2015, Jul 17 2014

[n: n in [1..10100000] | &+Intseq(n) eq 2]; // Vincenzo Librandi, Mar 07 2013

/* As a triangular array: */ [[10^n+10^m: m in [0..n]]: n in [0..8]]; // Bruno Berselli, Mar 07 2013

t = 10^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
With[{nn=7},Sort[Join[Table[FromDigits[PadRight[{2},n,0]],{n,nn}], FromDigits/@Flatten[Table[Table[Insert[PadRight[{1},n,0],1,i]],{n,nn},{i,2,n+1}],1]]]] (* Harvey P. Dale, Nov 15 2011 *)
Select[Range[10^9], Total[IntegerDigits[#]] == 2&] (* Vincenzo Librandi, Mar 07 2013 *)
T[n_,k_]:=10^(n-1)+10^(k-1); Table[T[n,k],{n,8},{k,n}]//Flatten (* Stefano Spezia, Nov 03 2023 *)

a(n)=my(d=(sqrtint(8*n)-1)\2,t=n-d*(d+1)/2-1); 10^d + 10^t \\ Charles R Greathouse IV, Dec 19 2016

from itertools import count, islice
def agen(): yield from (10**i + 10**j for i in count(0) for j in range(i+1))
print(list(islice(agen(), 34))) # Michael S. Branicky, May 15 2022

from math import isqrt
def A052216(n): return 10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

def A052216(n,k): return 10^(n-1) + 10^(k-1)
flatten([[A052216(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Feb 22 2024

T(n,k) = 10^(n-1) + 10^(k-1) with 1 <= k <= n.
a(n) = 3*A237424(n) - 1. - Reinhard Zumkeller, Jan 28 2015
a(n) = 10^A003056(n-1) + 10^A002262(n-1). - Chai Wah Wu, Apr 08 2025

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

A038464 Sums of 2 distinct powers of 3.

Original entry on oeis.org

4, 10, 12, 28, 30, 36, 82, 84, 90, 108, 244, 246, 252, 270, 324, 730, 732, 738, 756, 810, 972, 2188, 2190, 2196, 2214, 2268, 2430, 2916, 6562, 6564, 6570, 6588, 6642, 6804, 7290, 8748, 19684, 19686, 19692, 19710, 19764, 19926, 20412, 21870, 26244, 59050, 59052

Offset: 1

Olivier Gérard

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

Base-3 interpretation of A038444.

Cf. A000244, A038465, A038466, A038467, A038468, A038469.

Sort[Plus @@@ Subsets[3^Range[0, 10], {2}]] (* Amiram Eldar, Jul 13 2022 *)

from math import isqrt
def A038464(n): return 3**(m:=isqrt(n<<3)+1>>1)+3**(n-1-(m*(m-1)>>1)) # Chai Wah Wu, Apr 05 2025

Offset corrected by Amiram Eldar, Jul 13 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

A052216 Sums of two powers of 10.

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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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A038464 Sums of 2 distinct powers of 3.

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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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A038453 Sums of 11 distinct powers of 10.

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