A049445 -id:A049445 - cp's OEIS frontend

A331819 Positive numbers k such that -k is a negative negabinary-Niven number, i.e., divisible by the sum of digits of its negabinary representation (A027615).

2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 33, 34, 36, 39, 40, 42, 44, 48, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 77, 78, 80, 84, 90, 92, 96, 100, 102, 104, 108, 111, 112, 114, 115, 116, 120, 123, 124, 126, 128, 129, 130, 132, 135, 136, 138, 140

Offset: 1

Amiram Eldar, Jan 27 2020

			6 is a term since A039724(-6) = 1110 and 1 + 1 + 1 + 0 = 3 is a divisor of 6.

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

Cf. A005352, A027615, A039724, A005349, A049445, A328208, A328212, A331085, A331088, A331728.

negaBinWt[n_] := negaBinWt[n] = If[n==0, 0, negaBinWt[Quotient[n-1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; Select[Range[100], negaBinNivenQ]

A381581 Numbers divisible by the sum of the digits in their Chung-Graham representation (A381579).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 20, 21, 22, 24, 27, 28, 30, 40, 42, 44, 45, 48, 55, 56, 57, 58, 60, 66, 70, 72, 75, 76, 80, 84, 90, 92, 95, 96, 100, 102, 110, 111, 112, 115, 116, 120, 132, 135, 138, 140, 144, 150, 152, 153, 156, 168, 170, 175, 176, 180, 186, 190, 195, 198

Offset: 1

Amiram Eldar, Feb 28 2025

Numbers k such that A291711(k) divides k.
Analogous to Niven numbers (A005349) with the Chung-Graham representation (A381579) instead of the decimal representation.
A001906(k) = Fibonacci(2*k) is a term for all k >= 1.
If k is not divisible by 3 (A001651), then Fibonacci(2*k) + 1 is a term.

			4 is a term since A291711(4) = 1 divides 4.
6 is a term since A291711(6) = 2 divides 6.

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

Cf. A000045, A001651, A001906, A291711.
Subsequences: A381582, A381583, A381584, A381585.
Similar sequences: A005349, A049445, A064150, A328208, A328212.

f[n_] := f[n] = Fibonacci[2*n]; q[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; Divisible[n, s]]; Select[Range[200], q]

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
isok(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); !(n % s);}

A235602 a(n) = n/wt(n) if wt(n) divides n, otherwise a(n) = n, where wt(n) is the binary weight of n (A000120).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 6, 13, 14, 15, 16, 17, 9, 19, 10, 7, 22, 23, 12, 25, 26, 27, 28, 29, 30, 31, 32, 33, 17, 35, 18, 37, 38, 39, 20, 41, 14, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 11, 56, 57, 58, 59, 15, 61, 62, 63, 64, 65, 33, 67, 34, 23, 70, 71, 36, 73, 74, 75, 76, 77, 78, 79, 40, 27, 82

Offset: 1

N. J. A. Sloane, Jan 18 2014

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

Cf. A000079, A000120, A049445, A065878, A235600, A235601.

bw[n_]:=Module[{w=DigitCount[n,2,1]},If[Divisible[n,w],n/w,n]]; Array[ bw,90] (* Harvey P. Dale, Nov 06 2016 *)

a(n) = my(s=hammingweight(n)); if (n % s, n, n/s); \\ Michel Marcus, Jul 15 2021

From Amiram Eldar, Aug 04 2025: (Start)
a(n) = n if and only if n is in A065878 or A000079.
a(n) < n if and only if n is in A049445 but not in A000079. (End)

A364006 Wythoff-Niven numbers: numbers that are divisible by the number of 1's in their Wythoff representation.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 12, 15, 18, 20, 21, 24, 26, 28, 32, 35, 39, 40, 42, 45, 47, 51, 52, 54, 55, 56, 60, 68, 72, 76, 80, 84, 86, 88, 90, 91, 98, 100, 102, 105, 117, 120, 123, 125, 135, 136, 138, 141, 143, 144, 156, 164, 168, 172, 174, 176, 178, 180, 188, 192

Offset: 1

Amiram Eldar, Jul 01 2023

Numbers k such that A135818(k) | k.

Includes all the positive even-indexed Fibonacci numbers (A001906), since the Wythoff representation of Fibonacci(2*n), for n >= 1, is 1 followed by n-1 0's.

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

Cf. A135818, A189921.
Subsequences: A364007, A364008, A364009.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352508.

wnQ[n_] := (s = Total[w[n]]) > 0 && Divisible[n, s] (* using the function w[n] from A364005 *)

A364123 Stolarsky-Niven numbers: numbers that are divisible by the number of 1's in their Stolarsky representation (A364121).

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 14, 16, 20, 22, 24, 27, 30, 36, 38, 40, 42, 44, 48, 54, 56, 57, 60, 65, 69, 72, 75, 80, 84, 85, 90, 92, 96, 98, 100, 102, 104, 108, 112, 116, 120, 124, 126, 132, 136, 138, 145, 147, 150, 153, 155, 159, 160, 175, 180, 185, 190, 195, 196, 205

Offset: 1

Amiram Eldar, Jul 07 2023

Numbers k such that A200649(k) | k.

Fibonacci(k) + 1 is a term if k !== 3 (mod 6) (i.e., k is in A047263).

			4 is a term since its Stolarsky representation, A364121(4) = 10, has one 1 and 4 is divisible by 1.
6 is a term since its Stolarsky representation, A364121(6) = 101, has 2 1's and 6 is divisible by 2.

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

Cf. A047263, A200649, A364121.
Subsequences: A364124, A364125, A364126.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
stolNivQ[n_] := n > 1 && Divisible[n, Total[stol[n]]];
Select[Range[200], stolNivQ]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
isA364123(n) = n > 1 && !(n % vecsum(stol(n)));

A385483 Where records occur in A385482.

Original entry on oeis.org

1, 3, 7, 11, 13, 19, 103, 391, 1811, 3589, 5147, 6683, 21883, 46159, 64133, 149839, 151013, 318377, 650543, 1279211, 42559939, 43120271, 55201423, 198069181, 265237811, 929670011, 930260173, 1879562281, 3320654641, 5390681357, 52883996713, 78842843063, 250434427519

Offset: 1

Amiram Eldar, Jun 30 2025

Amiram Eldar, Table of n, a(n) for n = 1..40

Cf. A049445, A144364 (decimal analog), A385482, A385484 (record values), A385486.

f[n_] := Module[{m = n, k = 1}, While[!Divisible[m, DigitSum[m, 2]], m += n; k++]; k];
seq[lim_] := Module[{s = {}, fm = -1, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, i]], {i, 1, lim}]; s]; seq[10^4]

f(n) = {my(m = n, k = 1); while(m % hammingweight(m), m += n; k++); k;}
list(lim) = my(fm = -1, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(i, ", ")));

A385482(a(n)) = A385484(n).

A385486 Where records occur in A385485.

Original entry on oeis.org

1, 2, 12, 126, 252, 504, 2040, 4080, 16380, 32760, 65520, 524286, 1048572, 4194300, 8388600, 134217720, 268435440, 7516192740, 10737418230, 21474836460, 137438953440, 274877906880, 274877906940, 549755813880, 8796093022200, 8796093022206, 17592186044412

Offset: 1

Amiram Eldar, Jun 30 2025

Chai Wah Wu, Table of n, a(n) for n = 1..27

Cf. A049445, A144376 (decimal analog), A385483, A385485, A385487 (record values).

f[n_] := Module[{m = n, k = 1}, While[Divisible[m, DigitSum[m, 2]], m += 2*n; k += 2]; k];
seq[lim_] := Module[{s = {}, fm = -1, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, i]], {i, 1, lim}]; s]; seq[10^4]

f(n) = {my(m = n, k = 1); while(!(m % hammingweight(m)), m += 2*n; k += 2); k;}
list(lim) = my(fm = -1, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(i, ", ")));

A385485(a(n)) = A385487(n).

a(21)-a(27) from Chai Wah Wu, Jul 02 2025

A385487 Records in A385485.

Original entry on oeis.org

3, 7, 29, 65, 69, 81, 257, 259, 4097, 4113, 4129, 262145, 262149, 1048577, 1048583, 16777217, 16777219, 38347927, 214748365, 214748369, 4294967297, 4294967299, 68719476737, 68719476769, 1099511627777, 4398046511105, 4398046511109

Offset: 1

Amiram Eldar, Jun 30 2025

Chai Wah Wu, Table of n, a(n) for n = 1..27

Cf. A049445, A144375 (decimal analog), A385484, A385485, A385486 (indices of records).

f[n_] := Module[{m = n, k = 1}, While[Divisible[m, DigitSum[m, 2]], m += 2*n; k += 2]; k];
seq[lim_] := Module[{s = {}, fm = -1, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, fi]], {i, 1, lim}]; s]; seq[10^4]

f(n) = {my(m = n, k = 1); while(!(m % hammingweight(m)), m += 2*n; k += 2); k;}
list(lim) = my(fm = -1, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(fi, ", ")));

a(n) = A385485(A385486(n)).

a(21)-a(27) from Chai Wah Wu, Jul 02 2025

A180490 Numbers k such that (A000120(k))^2 divides k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 64, 68, 72, 80, 81, 96, 128, 132, 136, 144, 160, 162, 192, 240, 252, 256, 260, 261, 264, 272, 288, 320, 324, 368, 384, 425, 432, 464, 480, 504, 512, 516, 520, 522, 528, 544, 576, 624, 625, 637, 640, 648, 675, 688, 720

Offset: 1

Ctibor O. Zizka, Sep 08 2010

This is a subsequence of A049445.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000120, A049445.

fQ[n_] := Divisible[ n, DigitCount[n, 2, 1]^2]; Select[ Range@ 735, fQ] (* Robert G. Wilson v, Sep 10 2010 *)

isok(k) = !(k % hammingweight(k)^2); \\ Amiram Eldar, Jan 10 2025

def ok(n): return n and n%n.bit_count()**2 == 0
print([k for k in range(721) if ok(k)]) # Michael S. Branicky, Jan 10 2025

{k: (A000120(k))^2 | k}.

a(43) onwards from Robert G. Wilson v, Sep 10 2010

A306263 Numbers k such that, for any divisor d of k, the Hamming weight of d divides k.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 40, 42, 48, 60, 64, 66, 68, 72, 80, 84, 92, 96, 108, 116, 120, 126, 128, 132, 136, 144, 156, 160, 168, 172, 180, 184, 192, 204, 212, 216, 222, 228, 232, 240, 246, 252, 256, 264, 272, 276, 284, 288, 300, 310

Offset: 1

Rémy Sigrist, Mar 02 2019

The Hamming weight of a number is given by A000120.
This sequence is a binary variant of A285815.
This sequence is infinite as it contains all powers of 2 (A000079).
All terms belong to A049445.
If k belongs to the sequence, then 2*k belongs to the sequence.
All terms except 1 are even. - Robert Israel, Mar 05 2019

			For n = 108:
- the divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108,
- the corresponding Hamming weights are 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 4,
- they all divide 108,
- hence 108 belongs to the sequence.
For n = 98:
- the divisors of 98 are 1, 2, 7, 14, 49, 98,
- the correspond Hamming weights are 1, 1, 3, 3, 3, 3,
- 3 does not divide 98,
- hence 98 does not belong to the sequence.

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

Cf. A000079, A000120, A049445, A141586, A285815.

Positions of zeros in A324393.

[k:k in [1..310]| forall{d:d in Divisors(k)| k mod &+Intseq(d,2) eq 0}]; // Marius A. Burtea, Dec 30 2019

filter:= proc(n) local F;
  F:= map(convert,map(convert,numtheory:-divisors(n),base,2),`+`);
  andmap(t -> n mod t = 0, F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 05 2019

Select[Range@ 310, With[{k = #}, AllTrue[Divisors@ k, Mod[k, DigitCount[#, 2, 1]] == 0 &]] &] (* Michael De Vlieger, Mar 05 2019 *)

is(n) = fordiv(n,d,if (n%hammingweight(d), return (0))); return ( )

cp's OEIS Frontend

A331819 Positive numbers k such that -k is a negative negabinary-Niven number, i.e., divisible by the sum of digits of its negabinary representation (A027615).

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A381581 Numbers divisible by the sum of the digits in their Chung-Graham representation (A381579).

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A235602 a(n) = n/wt(n) if wt(n) divides n, otherwise a(n) = n, where wt(n) is the binary weight of n (A000120).

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A364006 Wythoff-Niven numbers: numbers that are divisible by the number of 1's in their Wythoff representation.

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A364123 Stolarsky-Niven numbers: numbers that are divisible by the number of 1's in their Stolarsky representation (A364121).

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A385483 Where records occur in A385482.

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A385486 Where records occur in A385485.

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A385487 Records in A385485.

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