A058898 -id:A058898 - cp's OEIS frontend

A003635 Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).

62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, 326, 371, 372, 377, 381, 383, 386, 387, 395, 411, 416, 422, 426, 431, 432, 438, 441, 443, 461, 466, 471, 476, 482, 483, 486, 488, 491, 492, 493, 494, 497, 498, 516, 521, 522, 527, 531, 533, 536

Offset: 1

N. J. A. Sloane and Mira Bernstein

J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Mondot, Table of n, a(n) for n = 1..10867
Giovanni Resta, Numbers Aplenty: Inconsummate numbers

Cf. A003634, A052489, A052490, A052491, A058898-A058917.

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

nmax = 1000; Reap[ Do[k = n; kmax = 100*n; While[ Tr[ IntegerDigits[k]]*n != k && k < kmax, k = k + n]; If[k == kmax, Sow[n]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Jul 12 2012 *)

from itertools import count, islice, combinations_with_replacement
def A003635_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 9*l*n < 10**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(10),l):
                if (s:=sum(d))>0 and sorted(str(s*n)) == [str(e) for e in d]:
                    break
            else:
                continue
            break
A003635_list = list(islice(A003635_gen(),20)) # Chai Wah Wu, May 09 2023

A052491 Smallest "inconsummate number" in base n: smallest number such that in base n, no number is this multiple of the sum of its digits.

Original entry on oeis.org

13, 17, 29, 16, 27, 30, 42, 46, 62, 68, 86, 92, 114, 122, 147, 154, 182, 192, 222, 232, 266, 278, 314, 326, 367, 380, 422, 436, 482, 498, 546, 562, 614, 632, 688, 704, 762, 782, 842, 862, 926, 948, 1014, 1036, 1107, 1130, 1202, 1226, 1302, 1328, 1406, 1432

Offset: 2

J. H. Conway, Dec 30 2000

			a(10) = 62, from A003635.

Chai Wah Wu, Table of n, a(n) for n = 2..174

Cf. A003635, A058898-A058907.

Do[n = 1; While[k = n; While[ Apply[ Plus, IntegerDigits[k, b] ]*n != k && k < 100n, k += n ]; k != 100n, n++ ]; Print[n], {b, 2, 54} ]

from itertools import count, combinations_with_replacement
from sympy.ntheory import digits
def A052491(n):
    for k in count(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

Corrected and extended by David Radcliffe, Jan 08 2001

More terms from David W. Wilson, Jan 10 2001

A058907 Inconsummate numbers in base 12: no number is this multiple of the sum of its digits (in base 12).

Original entry on oeis.org

86, 87, 88, 90, 99, 101, 102, 112, 113, 114, 125, 126, 138, 229, 235, 244, 245, 246, 256, 258, 269, 270, 282, 307, 373, 379, 385, 391, 392, 400, 402, 426, 451, 464, 530, 535, 536, 542, 543, 547, 548, 607, 608, 620, 667, 673, 674, 679, 680, 685

Offset: 1

N. J. A. Sloane, Jan 09 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..10100

Cf. A003635, A052491, A058898, A058899, A058900, A058901, A058902, A058903, A058904, A058905, A058906.

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

base=12; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Sep 23 2017; after N. J. A. Sloane in A058906 *)

from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058907_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 11*l*n < 12**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(12),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,12)[1:]) == list(d):
                    break
            else:
                continue
            break
A058907_list = list(islice(A058907_gen(),20)) # Chai Wah Wu, May 10 2023

A058906 Inconsummate numbers in base 11: no number is this multiple of the sum of its digits (in base 11).

Original entry on oeis.org

68, 70, 79, 80, 82, 92, 104, 200, 202, 212, 214, 224, 225, 248, 260, 314, 320, 332, 380, 392, 452, 458, 464, 490, 502, 508, 512, 513, 514, 518, 520, 524, 530, 562, 568, 574, 578, 579, 580, 584, 585, 590, 592, 595, 598, 599, 622, 628, 634, 635

Offset: 1

N. J. A. Sloane, Jan 09 2001

Cf. A003635, A052491, A058898-A058907.

digitsum := proc (n,b) local i; add(i,i=convert(n,base,b)) end; b := 11:N := 43922; L := []: for n from 1 to N do k := digitsum(n,b): if (n mod k)=0 then L := [op(L), n/k] fi: od: M := []: for i from 1 to 1000 do if not(member(i,L)) then M := [op(M),i] fi od: lprint(M);

base = 11; Do[k = n; While[ Apply[ Plus, IntegerDigits[k, base] ]*n != k && k < 250n, k += n]; If[k == 250n, Print[n] ], {n, 1, 10^3} ]

from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058906_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 10*l*n < 11**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(11),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,11)[1:]) == list(d):
                    break
            else:
                continue
            break
A058906_list = list(islice(A058906_gen(),20)) # Chai Wah Wu, May 09 2023

A065880 Largest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

Original entry on oeis.org

0, 1, 2, 6, 4, 10, 12, 21, 8, 18, 20, 55, 24, 0, 42, 60, 16, 34, 36, 0, 40, 126, 110, 115, 48, 0, 0, 108, 84, 116, 120, 155, 32, 66, 68, 0, 72, 222, 0, 156, 80, 246, 252, 172, 220, 180, 230, 0, 96, 0, 0, 204, 0, 318, 216, 0, 168, 285, 232, 295, 240, 366, 310, 378, 64, 130

Offset: 0

Henry Bottomley, Nov 26 2001

a(n) is bounded above by n*A272756(n), so a program only has to check values up to that point to see if a(n) is zero. - Peter Kagey, May 05 2016

			a(23)=115 since 115 is written in binary as 1110011 and 115/(1+1+1+0+0+1+1)=23 and there is no higher possibility (if k is more than 127 then k divided by its number of binary 1's is more than 26).

Peter Kagey, Table of n, a(n) for n = 0..10000

A052489 is the base 10 equivalent.

Cf. A000120, A049445, A058898, A065413, A065878, A065879, A272756.

Table[SelectFirst[Reverse@ Range@ #, First@ DigitCount[#, 2] == #/n &] &[n SelectFirst[Range[2^12], # > IntegerLength[n #, 2] &]], {n, 80}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, May 05 2016, Version 10.2 *)

A065413 Number of positive solutions to "numbers that are n times their number of binary 1's".

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 1, 0, 0, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 3, 0, 1, 0, 0, 1, 0, 2, 2, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 1, 1, 2, 1, 0, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2

Offset: 1

Henry Bottomley, Nov 23 2001

Equivalently, this is the number of ways to write n as an arithmetic mean of distinct powers of 2. [Brian Kell, Mar 01 2009]

			a(23)=3 since 69, 92 and 115 are written in binary as 1000101, 1011100 and 1110011 and 69=23*3, 92=23*4 and 115=23*5.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000120, A037478, A058898, A272797 (greedy inverse).

N:= 1000: # to get a(1) to a(N)
A:= Vector(N):
for x from 1 while x/(1+ilog2(x)) <= N do
  v:= x/convert(convert(x,base,2),`+`);
  if v::integer and v <= N then
    A[v]:= A[v]+1
  fi
od:
seq(A[i],i=1..N); # Robert Israel, May 06 2016
# alternative program
read("transforms") :
A065413 := proc(n)
    local bdgs,a,x;
    a := 0 ;
    for bdgs from 1 do
        x := n*bdgs ;
        # x must have bdgs bits set, so x =bdgs*n >= 2^bdgs-1.
        if n < (2^bdgs-1)/x then
            break;
        elif wt(x) = bdgs then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 11 2016

A065878 Numbers which are not an integer multiple of their number of binary 1's.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 65, 67, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97, 98, 99, 100

Offset: 1

Henry Bottomley, Nov 26 2001

			5 is in the sequence since 5 = 101_2 and 5 is not a multiple of 1 + 0 + 1 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Complement of A049445.
The base-10 equivalent is A065877.
Cf. A000120, A058898, A065413, A065879, A065880.

Select[Range[100],!IntegerQ[#/Total[IntegerDigits[#,2]]]&]  (* Harvey P. Dale, Apr 20 2011 *)

isok(k) = k % hammingweight(k); \\ Amiram Eldar, Aug 04 2025

A065879 a(n) is the smallest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 6, 4, 10, 12, 21, 8, 18, 20, 55, 24, 0, 42, 60, 16, 34, 36, 0, 40, 126, 110, 69, 48, 0, 0, 81, 84, 116, 120, 155, 32, 66, 68, 0, 72, 185, 0, 156, 80, 205, 252, 172, 220, 180, 138, 0, 96, 0, 0, 204, 0, 212, 162, 0, 168, 228, 232, 295, 240, 366, 310, 378, 64, 130

Offset: 1

Henry Bottomley, Nov 26 2001

a(n) is bounded above by n*A272756(n), so a program only has to check values up to that point to see if a(n) is zero. - Peter Kagey, May 05 2016

			a(23) is 69 since 69 is written in binary as 1000101, 69/(1+0+0+0+1+0+1)=23 and there is no smaller possibility (neither 23 nor 46 are divisible by their number of binary 1's).

Peter Kagey, Table of n, a(n) for n = 1..10000

A003634 is the base-10 equivalent.

Cf. A000120, A049445, A058898, A065413, A065878, A065880, A272756.

Table[SelectFirst[Range[2^12], # == n First@ DigitCount[#, 2] &] /. k_ /; MissingQ@ k -> 0, {n, 80}] (* Michael De Vlieger, May 05 2016, Version 10.2 *)

A058899 Inconsummate numbers in base 3: no number is this multiple of the sum of its digits (in base 3).

Original entry on oeis.org

17, 32, 44, 51, 94, 95, 96, 106, 107, 112, 118, 132, 148, 153, 199, 224, 226, 232, 235, 236, 238, 256, 265, 268, 269, 274, 277, 281, 282, 284, 285, 288, 296, 308, 318, 321, 334, 336, 343, 352, 354, 368, 396, 442, 443, 444, 454, 459, 469, 472

Offset: 1

N. J. A. Sloane, Jan 09 2001

Daniel Mondot, Table of n, a(n) for n = 1..13275

Cf. A003635, A052491, A058898-A058907.

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

base=3; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250 n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Sep 23 2017 *)

from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058899_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 2*l*n < 3**(l-1):
                yield n
                break
            for d in combinations_with_replacement((0,1,2),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,3)[1:]) == list(d):
                    break
            else:
                continue
            break
A058899_list = list(islice(A058899_gen(),20)) # Chai Wah Wu, May 10 2023

A058900 Inconsummate numbers in base 4: no number is this multiple of the sum of its digits (in base 4).

Original entry on oeis.org

29, 41, 71, 83, 93, 111, 113, 114, 116, 117, 122, 123, 125, 135, 137, 143, 146, 153, 164, 167, 191, 197, 201, 237, 242, 263, 275, 279, 282, 284, 285, 291, 303, 305, 311, 323, 326, 327, 332, 359, 362, 369, 372, 375, 377, 382, 383, 389, 407, 410

Offset: 1

N. J. A. Sloane, Jan 09 2001

Daniel Mondot, Table of n, a(n) for n = 1..15650

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 A058900_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 3*l*n < 1<<((l-1)<<1):
                yield n
                break
            for d in combinations_with_replacement((0,1,2,3),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,4)[1:]) == list(d):
                    break
            else:
                continue
            break
A058900_list = list(islice(A058900_gen(),20)) # Chai Wah Wu, May 10 2023

cp's OEIS Frontend

A003635 Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).

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A052491 Smallest "inconsummate number" in base n: smallest number such that in base n, no number is this multiple of the sum of its digits.

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A058907 Inconsummate numbers in base 12: no number is this multiple of the sum of its digits (in base 12).

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A058906 Inconsummate numbers in base 11: no number is this multiple of the sum of its digits (in base 11).

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A065880 Largest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

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A065413 Number of positive solutions to "numbers that are n times their number of binary 1's".

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A065878 Numbers which are not an integer multiple of their number of binary 1's.

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A065879 a(n) is the smallest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

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