A052491 -id:A052491 - cp's OEIS frontend

A058905 Inconsummate numbers in base 9: no number is this multiple of the sum of its digits (in base 9).

46, 47, 48, 56, 58, 66, 76, 86, 136, 138, 167, 176, 222, 227, 228, 248, 258, 298, 302, 308, 312, 316, 318, 338, 343, 344, 347, 348, 352, 354, 356, 358, 362, 374, 383, 384, 392, 398, 402, 403, 404, 406, 407, 408, 411, 412, 414, 416, 422, 423

Offset: 1

N. J. A. Sloane, Jan 09 2001

Cf. A003635, A052491, A058898-A058907.

Maple
```
For Maple code see A058906.
```

from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058905_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if l*n<<3 < 9**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(9),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,9)[1:]) == list(d):
                    break
            else:
                continue
            break
A058905_list = list(islice(A058905_gen(),20)) # Chai Wah Wu, May 10 2023

A061381 Smallest "inconsummate number" in base n greater than in the previous base.

Original entry on oeis.org

13, 17, 29, 46, 64, 86, 105, 136, 161, 200, 229, 276, 309, 362, 419, 460, 505, 572, 621, 694, 749, 830, 889, 978, 1054, 1136, 1205, 1306, 1381, 1490, 1569, 1684, 1769, 1892, 1999, 2112, 2205, 2342, 2441, 2584, 2689, 2840, 2949, 3106, 3269, 3386, 3505, 3678

Offset: 2

Robert G. Wilson v, Jun 08 2001

Chai Wah Wu, Table of n, a(n) for n = 2..226

Cf. A052491.

n = 1; Do[ While[k = n; While[ Apply[ Plus, IntegerDigits[k, b] ]*n != k && k < 100n, k += n]; k != 100n, n++ ]; Print[n], {b, 2, 60} ]

from functools import lru_cache
from itertools import count, combinations_with_replacement
from sympy.ntheory import digits
@lru_cache(maxsize=None)
def A061381(n):
    for k in count((0 if n <= 2 else A061381(n-1))+1):
        for l in count(1):
            if (n-1)*l*k < n**(l-1):
                return k
            for d in combinations_with_replacement(range(n),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*k,n)[1:]) == list(d):
                    break
            else:
                continue
            break # Chai Wah Wu, May 09 2023

A137225 Triangle T(k,q) of minimal q-Niven numbers: smallest number such that the sum of its digits in base q equals k, 2<=q<=k+1.

Original entry on oeis.org

1, 3, 2, 7, 5, 3, 15, 8, 7, 4, 31, 17, 11, 9, 5, 63, 26, 15, 14, 11, 6, 127, 53, 31, 19, 17, 13, 7, 255, 80, 47, 24, 23, 20, 15, 8, 511, 161, 63, 49, 29, 27, 23, 17, 9, 1023, 242, 127, 74, 35, 34, 31, 26, 19, 10, 2047, 485, 191, 99, 71, 41, 39, 35, 29, 21, 11, 4095, 728, 255

Offset: 1

R. J. Mathar, Mar 07 2008

			T(8,4) =47 because 47, written 233 in base q=4, is the smallest number with
digit sum 2+3+3=8=k in base q=4. The triangle reads T(k,q), k=1,2,...,
2<=q up to the diagonal, after which the values stay constant:
1 1 1 1 1 1 1 1 1
3 2 2 2 2 2 2 2 2
7 5 3 3 3 3 3 3 3
15 8 7 4 4 4 4 4 4
31 17 11 9 5 5 5 5 5
63 26 15 14 11 6 6 6 6
127 53 31 19 17 13 7 7 7
255 80 47 24 23 20 15 8 8
511 161 63 49 29 27 23 17 9
1023 242 127 74 35 34 31 26 19
...

H. Fredricksen, E. J. Ionascu, F. Luca, P. Stanica, Minimal Niven numbers, arXiv:0803.0477 [math.NT]

Cf. A005349, A052491.

sd := proc(n,b) local i ; add(i,i=convert(n,base,b)) ; end: T := proc(k,q) local a; for a from 1 do if sd(a,q) = k then RETURN(a) ; fi ; od: end: for k from 1 to 20 do for q from 2 to k+1 do printf("%d, ",T(k,q)) ; od: od:

T(k,2)=A000225(k). T(k,k+1)=2k-1. Conjecture: T(k,3)=A062318(k), verified up to k=23.

cp's OEIS Frontend

A058905 Inconsummate numbers in base 9: no number is this multiple of the sum of its digits (in base 9).

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A061381 Smallest "inconsummate number" in base n greater than in the previous base.

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Mathematica

Python

A137225 Triangle T(k,q) of minimal q-Niven numbers: smallest number such that the sum of its digits in base q equals k, 2<=q<=k+1.

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