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

Author

Henry Bottomley, Feb 01 2000

Keywords

• easy
• nonn
• tabl

Comments

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

Examples

			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)
		

Links

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

Crossrefs

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.

Programs

Haskell

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

Magma

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

Magma

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

Mathematica

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 *)

PARI

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

Python

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

Python

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

SageMath

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

Formula

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