A049445 -id:A049445 - cp's OEIS frontend

A199238 n mod (number of ones in binary representation of n).

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 0, 1, 0, 0, 1, 3, 0, 1, 2, 3, 1, 1, 2, 1, 0, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 3, 2, 1, 2, 2, 0, 1, 2, 3, 1, 1, 2, 0, 2, 1, 2, 4, 0, 1, 2, 3, 0, 1, 0, 1, 0, 0, 1, 3, 0, 1, 2, 3, 1, 1, 2, 4, 0, 0, 1, 3, 0, 1, 2

Offset: 1

Reinhard Zumkeller, Nov 04 2011

nonn

a(A049445(n)) = 0;
a(n) = n - A161764(n);
a(A199262(n)) = n and a(m) <> n for m < A199262(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A000120, A070635, A049445, A161764, A199262.

a199238 n = a199238_list !! (n-1)
a199238_list = zipWith mod [1..] $ tail a000120_list

Mod[#,DigitCount[#,2,1]]&/@Range[90] (* Harvey P. Dale, Nov 08 2011 *)

A199238(n)=n%norml2(binary(n))  \\ M. F. Hasler, Oct 09 2012

A352342 Lazy-Pell-Niven numbers: numbers that are divisible by the sum of the digits in their maximal (or lazy) representation in terms of the Pell numbers (A352339).

Original entry on oeis.org

1, 2, 4, 9, 12, 15, 20, 24, 25, 28, 30, 35, 40, 48, 50, 54, 56, 60, 63, 64, 70, 72, 78, 84, 88, 91, 96, 102, 115, 120, 136, 144, 160, 162, 168, 180, 182, 184, 189, 207, 209, 210, 216, 217, 234, 246, 256, 261, 270, 304, 306, 308, 315, 320, 328, 333, 350, 352, 357

Offset: 1

Amiram Eldar, Mar 12 2022

Numbers k such that A352340(k) | k.

			4 is a term since its maximal Pell representation, A352339(4) = 11, has the sum of digits A352340(4) = 1+1 = 2 and 4 is divisible by 2.

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

Cf. A352339, A352340.
Subsequences: A352343, A352344, A352345.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; q[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; Divisible[n, Plus @@ v[[i[[1, 1]] ;; -1]]]]; Select[Range[300], q]

A352508 Catalan-Niven numbers: numbers that are divisible by the sum of the digits in their representation in terms of the Catalan numbers (A014418).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 14, 16, 18, 21, 24, 28, 30, 32, 33, 40, 42, 44, 45, 48, 55, 56, 57, 60, 65, 72, 78, 80, 84, 88, 95, 100, 105, 112, 126, 128, 130, 132, 134, 135, 138, 140, 144, 145, 146, 147, 152, 155, 156, 168, 170, 174, 180, 184, 185, 195, 210, 216

Offset: 1

Amiram Eldar, Mar 19 2022

Numbers k such that A014420(k) | k.
All the Catalan numbers (A000108) are terms.
If k is an odd Catalan number (A038003), then k+1 is a term.

			4 is a term since its Catalan representation, A014418(4) = 20, has the sum of digits A014420(4) = 2 + 0 = 2 and 4 is divisible by 2.
9 is a term since its Catalan representation, A014418(9) = 120, has the sum of digits A014420(9) = 1 + 2 + 0 = 3 and 9 is divisible by 3.

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

Cf. A014418, A014420.
Subsequences: A000108, A038003, A352509, A352510, A352511.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342.

c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; Select[Range[216], q]

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

A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 16, 20, 22, 24, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 68, 72, 75, 76, 84, 86, 87, 88, 92, 93, 95, 96, 99, 100, 104, 105, 108, 112, 115, 117, 120, 125, 126, 128, 129, 132, 135, 136, 138, 140

Offset: 1

Amiram Eldar, Jul 14 2023

Numbers k such that A364215(k) | k.

A007583 is a subsequence since A364215(A007583(n)) = 1 for n >= 0.

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

Cf. A280049, A364215.
Subsequences: A007583, A364217, A364218, A364219, A364220, A364221.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

seq[kmax_] := Module[{m = 1, s = {}}, Do[If[Divisible[k, DigitCount[m, 2, 1]], AppendTo[s, k]]; While[m++; OddQ[IntegerExponent[m, 2]]], {k, 1, kmax}]; s]; seq[140]

lista(kmax) = {my(m = 1); for(k = 1, kmax, if( !(k % sumdigits(m, 2)), print1(k,", ")); until(valuation(m, 2)%2 == 0, m++));}

A356552 a(n) is the least base b > 1 where the sum of digits of n divides n.

Original entry on oeis.org

2, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 7, 3, 2, 17, 2, 19, 2, 2, 11, 23, 2, 3, 5, 3, 3, 29, 3, 31, 2, 3, 2, 3, 2, 37, 19, 3, 2, 41, 2, 43, 6, 3, 23, 47, 2, 7, 4, 5, 4, 53, 3, 2, 3, 3, 29, 59, 2, 61, 31, 3, 2, 3, 2, 67, 2, 2, 6, 71, 2, 73, 37, 3, 4, 3, 3, 79

Offset: 1

Rémy Sigrist, Aug 12 2022

This sequence is well defined: a(1) = 2, and for n > 1, the sum of digits of n in base n equals 1, which divides n.

See A356553 for the corresponding sum of digits.

			For n = 14:
- we have:
      b   sum of digits  divides 14?
      --  -------------  -----------
       2              3  no
       3              4  no
       4              5  no
       5              6  no
       6              4  no
       7              2  yes
- so a(14) = 7.

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

Cf. A049445, A356553.

a[n_] := Module[{b = 2}, While[!Divisible[n, Plus @@ IntegerDigits[n, b]], b++]; b]; Array[a, 100] (* Amiram Eldar, Aug 15 2022 *)

a(n) = { for (b=2, oo, if (n % sumdigits(n, b)==0, return (b))) }

from sympy.ntheory import digits
def a(n):
    b = 2
    while n != 0 and n%sum(digits(n, b)[1:]): b += 1
    return b
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Aug 12 2022

a(n) = 2 iff n belongs to A049445.

a(n) = n iff n is prime.

A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 20, 21, 22, 24, 26, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 64, 68, 69, 72, 75, 76, 80, 84, 85, 86, 87, 88, 90, 92, 93, 96, 99, 100, 104, 105, 106, 108, 111, 112, 115, 116, 117, 120

Offset: 1

Amiram Eldar, Jul 21 2023

Numbers k such that A265745(k) | k.

The positive Jacobsthal numbers, A001045(n) for n >= 1, are terms since their representation in Jacobsthal greedy base is one 1 followed by n-1 0's, so A265745(A001045(n)) = 1 divides A001045(n).

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

Cf. A265745, A265747.
Subsequences: A364380, A364381, A364382, A364383.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508, A364216.

greedyJacobNivenQ[n_] := Divisible[n, A265745[n]]; Select[Range[120], greedyJacobNivenQ] (* using A265745[n] *)

isA364379(n) = !(n % A265745(n)); \\ using A265745(n)

A376615 a(n) is the number of iterations that n requires to reach a noninteger under the map x -> x / wt(x), where wt(k) is the binary weight of k (A000120); a(n) = 0 if n is a power of 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 1, 1, 1, 0, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 30 2024

The powers of 2 are fixed points of the map, since wt(2^k) = 1 for all k >= 0. Therefore they are arbitrarily assigned the value a(2^k) = 0.

Each number n starts a chain of a(n) integers: n, n/wt(n), (n/wt(n))/wt(n/wt(n)), ..., of them the first a(n)-1 integers are binary Niven numbers (A049445).

			a(6) = 2 since 6/wt(6) = 3 and 3/wt(3) = 3/2 is a noninteger that is reached after 2 iterations.
a(20) = 3 since 20/wt(20) = 10, 10/wt(10) = 5 and 5/wt(5) = 5/2 is a noninteger that is reached after 3 iterations.

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

Cf. A000005, A000079, A000120, A049445, A065878, A376616, A376617, A376619.

a[n_] := a[n] = Module[{bw = DigitCount[n, 2, 1]}, If[bw == 1, 0, If[!Divisible[n, bw], 1, 1 + a[n/bw]]]]; Array[a, 100]

a(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + a(n/w)));}

a(n) = 0 if and only if n is in A000079 (by definition).
a(n) = 1 if and only if n is in A065878.
a(n) >= 2 if and only if n is in A049445 \ A000079 (i.e., n is a binary Niven number that is not a power of 2).
a(n) >= 3 if and only if n is in A376616 \ A000079.
a(n) >= 4 if and only if n is in A376617 \ A000079.
a(2*n) >= a(n).
a(3*2^n) = n+1 for n >= 0.
a(n) < A000005(n).

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 *)

A226169 Niven numbers when expressed in bases 1 through 10.

Original entry on oeis.org

1, 2, 4, 6, 24, 40, 48, 72, 120, 144, 180, 216, 252, 288, 324, 336, 360, 432, 504, 576, 648, 720, 756, 780, 840, 960, 1008, 1056, 1080, 1092, 1200, 1260, 1296, 1344, 1380, 1440, 1512, 1584, 1620, 1680, 1728, 1764, 1800, 1944, 2016, 2196, 2304, 2352, 2448

Offset: 1

Sergio Pimentel, May 29 2013

The first 10 odd terms greater than 1 are a(1151) = 543375, 5329233, 18640125, 19178775, 23186625, 30131535, 35026425, 36797775, 46101825, 51856875. - Giovanni Resta, Jun 01 2013

			Example: 336 is in the sequence because the sum of digits of 336 when expressed in bases 1 through 10 is: 336, 3, 4, 3, 8, 6, 12, 7, 8, 12; and 336 is divisible by all these numbers.  In this particular example 336 keeps this property in bases 11, 12 and 13, but not 14.

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

Cf. A049445, A118363, A005349, A144261, A144262.

Select[Range[10^4], Catch[Do[If[Mod[#, Total@IntegerDigits[#, b]] > 0, Throw@ False], {b, 2, 10}]; True] &] (* Giovanni Resta, May 29 2013 *)
t = Table[b = 2; While[s = Total[IntegerDigits[n, b]]; s < n && Mod[n, s] == 0, b++]; If[s == n, b = 0]; b, {n, 2000}]; Flatten[Position[t, ?(# == 0 || # > 10 &)]] (* _T. D. Noe, May 30 2013 *)

Missing a(17) and a(35)-a(49) from Giovanni Resta, May 29 2013

cp's OEIS Frontend

A199238 n mod (number of ones in binary representation of n).

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A352342 Lazy-Pell-Niven numbers: numbers that are divisible by the sum of the digits in their maximal (or lazy) representation in terms of the Pell numbers (A352339).

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A352508 Catalan-Niven numbers: numbers that are divisible by the sum of the digits in their representation in terms of the Catalan numbers (A014418).

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

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A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

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A356552 a(n) is the least base b > 1 where the sum of digits of n divides n.

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A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

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A376615 a(n) is the number of iterations that n requires to reach a noninteger under the map x -> x / wt(x), where wt(k) is the binary weight of k (A000120); a(n) = 0 if n is a power of 2.

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