A052224 Numbers whose sum of digits is 10.

19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 505, 514, 523, 532, 541, 550, 604, 613, 622, 631, 640

Offset: 1

Author

Henry Bottomley, Feb 01 2000

Keywords

• nonn
• base
• easy

Comments

Proper subsequence of A017173. - Rick L. Shepherd, Jan 12 2009

Subsequence of A227793. - Michel Marcus, Sep 23 2013

A007953(a(n)) = 10; number of repdigits = #{55,22222,1^10} = A242627(10) = 3. - Reinhard Zumkeller, Jul 17 2014

a(n) = A094677(n) for n = 1..28. - Reinhard Zumkeller, Nov 08 2015

The number of terms having <= m digits is the coefficient of x^10 in sum(i=0,9,x^i)^m = ((1-x^10)/(1-x))^m. - David A. Corneth, Jun 04 2016

In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 3921 terms from T. D. Noe)

Crossrefs

Cf. A007953, A017173, A228915, A254524.

Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).

Cf. A242614, A242627.

Cf. A094677.

Cf. A018900, A187813.

Sum of base-b digits equal b: A226636 (b = 3), A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9).

Programs

Haskell

a052224 n = a052224_list !! (n-1)
a052224_list = filter ((== 10) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

Magma

[n: n in [1..1000] | &+Intseq(n) eq 10 ]; // Vincenzo Librandi, Mar 10 2013

Maple

sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if sd(n) = 10 then n else end if end proc: seq(a(n), n = 1 .. 800); # Emeric Deutsch, Jan 16 2009

Mathematica

Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 7]], {s, Rest[IntegerPartitions[10]]}]]] (* T. D. Noe, Mar 08 2013 *)
Select[Range[1000], Total[IntegerDigits[#]] == 10 &] (* Vincenzo Librandi, Mar 10 2013 *)

PARI

isok(n) = sumdigits(n) == 10; \\ Michel Marcus, Dec 28 2015

PARI

\\ This algorithm needs a modified binomial.
C(n, k)=if(n>=k, binomial(n, k), 0)
\\ ways to roll s-q with q dice having sides 0 through n - 1.
b(s, q, n)=if(s<=q*(n-1), s+=q; sum(i=0, q-1, (-1)^i*C(q, i)*C(s-1-n*i, q-1)), 0)
\\ main algorithm; this program applies to all sequences of the form "Numbers whose sum of digits is m."
a(n,{m=10}) = {my(q); q = 2; while(b(m, q, 10) < n, q++); q--; s = m; os = m; r=0; while(q, if(b(s, q, 10) < n, n-=b(s, q, 10); s--, r+=(os-s)*10^(q); os = s; q--)); r+= s; r}
\\ David A. Corneth, Jun 05 2016

Python

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits, b=10, sod=10): # works for any base, sum-of-digits
    alst = [sod] if 0 <= sod < b else []
    nzdigs = [i for i in range(1, b) if i <= sod]
    nzmultiset = []
    for d in range(1, b):
        nzmultiset += [d]*(sod//d)
    for d in range(2, maxdigits + 1):
        fullmultiset = [0]*(d-1-(sod-1)//(b-1)) + nzmultiset
        for firstdig in nzdigs:
            target_sum, restmultiset = sod - int(firstdig), fullmultiset[:]
            restmultiset.remove(firstdig)
            for p in multiset_permutations(restmultiset, d-1):
              if sum(p) == target_sum:
                  alst.append(int("".join(map(str, [firstdig]+p)), b))
                  if p[0] == target_sum:
                      break
    return alst
print(auptodigs(4)) # Michael S. Branicky, Sep 14 2021

Python

def A052224(N = 19):
    """Return a generator of the sequence of all integers >= N with the same
    digit sum as N."""
    while True:
        yield N
        N = A228915(N) # skip to next larger integer with the same digit sum
a = A052224(); [next(a) for  in range(50)] # _M. F. Hasler, Mar 16 2022

Formula

a(n+1) = A228915(a(n)) for any n > 0. - Rémy Sigrist, Jul 10 2018

Extensions

Incorrect formula deleted by N. J. A. Sloane, Jan 15 2009

Extended by Emeric Deutsch, Jan 16 2009

Offset changed by Bruno Berselli, Mar 07 2013