A049445 -id:A049445 - cp's OEIS frontend

A330931 Numbers k such that both k and k + 1 are Niven numbers in base 2 (A049445).

1, 20, 68, 80, 115, 155, 184, 204, 260, 272, 284, 320, 344, 355, 395, 404, 424, 464, 555, 564, 595, 623, 624, 636, 664, 675, 804, 835, 846, 847, 864, 875, 888, 904, 972, 1028, 1040, 1075, 1088, 1124, 1164, 1182, 1211, 1224, 1239, 1266, 1280, 1304, 1315, 1424

Offset: 1

Amiram Eldar, Jan 03 2020

Cai proved that there are infinitely many runs of 4 consecutive Niven numbers in base 2. Therefore this sequence is infinite.

			20 is a term since 20 and 20 + 1 = 21 are both Niven numbers in base 2.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 382.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tianxin Cai, On 2-Niven numbers and 3-Niven numbers, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 118-120.
Wikipedia, Harshad number.
Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.

Cf. A049445, A328205, A328209, A328213, A330713, A330927, A330932, A330933.

f:=func; a:=[]; for k in [1..1500] do  if forall{m:m in [0..1]|f(k+m)} then Append(~a,k); end if; end for; a; // Marius A. Burtea, Jan 03 2020

binNivenQ[n_] := Divisible[n, Total @ IntegerDigits[n, 2]]; bnq1 = binNivenQ[1]; seq = {}; Do[bnq2 = binNivenQ[k]; If[bnq1 && bnq2, AppendTo[seq, k - 1]]; bnq1 = bnq2, {k, 2, 10^4}]; seq

def sbd(n): return sum(map(int, str(bin(n)[2:])))
def niv2(n): return n%sbd(n) == 0
def aupto(nn): return [k for k in range(1, nn+1) if niv2(k) and niv2(k+1)]
print(aupto(1424)) # Michael S. Branicky, Jan 20 2021

A330932 Starts of runs of 3 consecutive Niven numbers in base 2 (A049445).

Original entry on oeis.org

623, 846, 2358, 4206, 4878, 6127, 6222, 6223, 12438, 16974, 21006, 27070, 31295, 33102, 33103, 35343, 37134, 37630, 37638, 40703, 43263, 45550, 48190, 49230, 52590, 53262, 53263, 56110, 59630, 66198, 66702, 66703, 67878, 69310, 69487, 72655, 74766, 77230, 77958

Offset: 1

Amiram Eldar, Jan 03 2020

Cai proved that there are infinitely many runs of 4 consecutive Niven numbers in base 2. Therefore this sequence is infinite.

			623 is a term since 623, 624 and 625 are all Niven numbers in base 2.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 382.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tianxin Cai, On 2-Niven numbers and 3-Niven numbers, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 118-120.
Wikipedia, Harshad number.
Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.

Cf. A049445, A154701, A328210, A328214, A330931, A330933.

f:=func; a:=[]; for k in [1..80000] do  if forall{m:m in [0..2]|f(k+m)} then Append(~a,k); end if; end for; a; // Marius A. Burtea, Jan 03 2020

binNivenQ[n_] := Divisible[n, Total @ IntegerDigits[n, 2]]; bin = binNivenQ /@ Range[3]; seq = {}; Do[bin = Join[Rest[bin], {binNivenQ[k]}]; If[And @@ bin, AppendTo[seq, k - 2]], {k, 3, 8*10^4}]; seq

A330933 Starts of runs of 4 consecutive Niven numbers in base 2 (A049445).

Original entry on oeis.org

6222, 33102, 53262, 66702, 94830, 221550, 268302, 284910, 295182, 300750, 316590, 364110, 379950, 427470, 533950, 554190, 570030, 590862, 617550, 633390, 696750, 791790, 807630, 855150, 870990, 902670, 934350, 1081422, 1140270, 1282830, 1314510, 1330350, 1343502

Offset: 1

Amiram Eldar, Jan 03 2020

Cai proved that there are infinitely many runs of 4 consecutive Niven numbers in base 2.

Grundman proved that there are no runs of 5 or more consecutive Niven numbers in base 2.

			6222 is a term since 6222, 6223, 6224 and 6225 are all Niven numbers in base 2.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 382.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tianxin Cai, On 2-Niven numbers and 3-Niven numbers, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 118-120.
Helen G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.
Wikipedia, Harshad number.
Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.

Cf. A049445, A141769, A328211, A328215, A330931, A330932.

f:=func; a:=[]; for k in [1..1400000] do  if forall{m:m in [0..3]|f(k+m)} then Append(~a,k); end if; end for; a; // Marius A. Burtea, Jan 03 2020

binNivenQ[n_] := Divisible[n, Total @ IntegerDigits[n, 2]]; bin = binNivenQ /@ Range[4]; seq = {}; Do[bin = Join[Rest[bin], {binNivenQ[k]}]; If[And @@ bin, AppendTo[seq, k - 3]], {k, 4, 10^6}]; seq

A376616 Binary Niven numbers (A049445) k such that k/wt(k) is also a binary Niven number, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 32, 36, 40, 48, 64, 68, 72, 80, 96, 126, 128, 132, 136, 144, 160, 192, 240, 252, 256, 260, 264, 272, 276, 288, 320, 324, 345, 368, 384, 405, 414, 432, 460, 464, 480, 486, 504, 512, 516, 520, 528, 544, 552, 576, 624, 640, 648, 688, 690

Offset: 1

Amiram Eldar, Sep 30 2024

Numbers k such that A376615(k) = 0 or A376615(k) >= 3.

If m is a term then 2^k * m is a term for all k >= 0.

			12 is a term since 12/wt(12) = 6 is an integer and also 6/wt(6) = 3 is an integer.

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

Subsequence of A049445.
Subsequences: A000079, A376617, A376618.
Cf. A000120, A376615.

q[k_] := Module[{w = DigitCount[k, 2, 1]}, Divisible[k, w] && Divisible[k/w, DigitCount[k/w, 2, 1]]]; Select[Range[1000], q]

is(k) = {my(w = hammingweight(k)); !(k % w) && !((k/w) % hammingweight(k/w));}

A376617 Binary Niven numbers (A049445) k such that m = k/wt(k) and m/wt(m) are also binary Niven numbers, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 32, 40, 48, 64, 72, 80, 96, 128, 136, 144, 160, 192, 256, 264, 272, 288, 320, 384, 512, 520, 528, 544, 576, 640, 756, 768, 960, 1024, 1032, 1040, 1056, 1088, 1104, 1152, 1280, 1296, 1380, 1472, 1512, 1536, 1620, 1656, 1728, 1840, 1856, 1920, 1944

Offset: 1

Amiram Eldar, Sep 30 2024

Numbers k such that A376615(k) = 0 or A376615(k) >= 4.

			24 is a term since 24/wt(24) = 12 is an integer, 12/wt(12) = 6 is an integer, and 6/wt(6) = 3 is an integer.

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

Subsequence of A049445 and A376616.
A000079 is a subsequence.
Cf. A000120, A376615.

q[k_] := Module[{w = DigitCount[k, 2, 1], w2, m, n}, IntegerQ[m = k/w] && Divisible[m, w2 = DigitCount[m, 2, 1]] && Divisible[n = m/w2, DigitCount[n, 2, 1]]]; Select[Range[2000], q]

s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
is(k) = {my(sk = s(k)); sk == 0 || sk >= 4;}

A144302 Odd members of A049445.

Original entry on oeis.org

1, 21, 55, 69, 81, 115, 155, 185, 205, 261, 273, 285, 295, 321, 345, 355, 395, 405, 425, 465, 535, 555, 565, 595, 623, 625, 637, 665, 675, 749, 775, 805, 835, 847, 861, 865, 875, 889, 905, 973, 1001, 1029, 1041, 1075, 1089, 1095, 1125, 1165, 1183, 1211, 1225, 1239

Offset: 1

Roger L. Bagula, Dec 07 2008

nonn

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

Intersection of A005408 and A049445.

Cf. A005349, A152567.

Select[Range[1, 1240, 2], Divisible[#, DigitCount[#, 2, 1]] &] (* Amiram Eldar, Aug 16 2020 *)

Edited by N. J. A. Sloane, Dec 07 2008

A363787 Primitive binary Niven numbers: binary Niven numbers (A049445) that are not twice another binary Niven number.

Original entry on oeis.org

1, 6, 10, 18, 21, 34, 55, 60, 66, 69, 81, 92, 108, 115, 116, 126, 130, 155, 156, 172, 180, 185, 204, 205, 212, 222, 228, 246, 258, 261, 273, 284, 285, 295, 300, 308, 318, 321, 332, 340, 345, 355, 356, 366, 378, 395, 396, 404, 405, 414, 420, 425, 438, 452, 462

Offset: 1

Amiram Eldar, Jun 22 2023

Every binary Niven number is of the form m*2^k, where m is a term of this sequence and k >= 0.
Includes all the odd binary Niven numbers (A144302).
This sequence is infinite. E.g., 16^k + 4^k + 1 is a term for all k >= 1.

			6 is a term as 6 is a binary Niven number and 6/2 = 3 is not a binary Niven number.

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

Subsequence of A049445.
Disjoint union of A144302 and A363788.
A363789 is a subsequence.
Cf. A356349 (decimal analog).

binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; q[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]); Select[Range[500], q]

isbinniv(n) = !(n % hammingweight(n));
is(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));

A337077 Binary Niven numbers (A049445) with a record gap to the next binary Niven number.

Original entry on oeis.org

1, 2, 12, 24, 96, 690, 1386, 3024, 3738, 3794, 5544, 22834, 57278, 68908, 89060, 196240, 360000, 388421, 524160, 1556360, 1572480, 2359140, 3929940, 8057711, 11484900, 15201585, 16115505, 19910436, 32444160, 7348411575, 16097143458, 33273395232, 51333952011

Offset: 1

Amiram Eldar, Aug 14 2020

The corresponding record gaps are 1, 2, 4, 8, 12, 18, 26, 27, 33, 38, 42, 44, 46, 50, 58, 68, 74, 77, 103, 109, 122, 137, 156, 157, 165, 189, 191, 204, 240, 265, 267, 312, 333, ...

De Koninck, Doyon and Kátai (2003) proved that the asymptotic density of the Niven numbers in any base >= 2 is 0. Therefore, the asymptotic density of the binary Niven numbers is 0 and this sequence is infinite.

			The first 8 binary Niven numbers are 1, 2, 4, 6, 8, 10, 12 and 16. The differences between them are 1, 2, 2, 2, 2, 2 and 4. The record gaps, 1, 2 and 4, occur at 1, 2 and 12.

Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
Jean-Marie De Koninck, Nicolas Doyon and Imre Kátai, On the counting function for the Niven numbers, Acta Arithmetica, Vol. 106, No. 3 (2003), 265-275.

Cf. A049445, A330931, A337076.

binNivenQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; gapmax = 0; n1 = 1; s = {}; Do[If[binNivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^6}]; s

A385482 a(n) is the least number k such that k*n is a binary Niven number (A049445).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 5, 1, 10, 3, 4, 1, 2, 1, 12, 1, 1, 3, 3, 1, 12, 5, 3, 3, 4, 2, 5, 1, 2, 1, 12, 1, 5, 6, 4, 1, 5, 1, 4, 3, 4, 2, 12, 1, 12, 6, 4, 3, 4, 2, 1, 3, 4, 2, 5, 1, 6, 5, 2, 1, 2, 1, 12, 1, 1, 6, 4, 1, 6, 3, 4, 3, 4, 2, 5, 1, 1, 3, 4, 1, 4

Offset: 1

Amiram Eldar, Jun 30 2025

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

Cf. A049445, A144261 (decimal analog), A363788, A385483 (indices of records), A385484 (record values), A385485.

a[n_] := Module[{m = n, k = 1}, While[!Divisible[m, DigitSum[m, 2]], m += n; k++]; k]; Array[a, 100]

a(n) = {my(m = n, k = 1); while(m % hammingweight(m), m += n; k++); k;}

from itertools import count
def a(n): return next(k for k in count(1) if (m:=k*n)%m.bit_count() == 0)
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Jun 30 2025

a(n) = 1 if and only if n is in A049445.

a(n) = 2 if and only if 2*n is in A363788.

A385485 a(n) is the least number k such that k*n is not a binary Niven number (A049445).

Original entry on oeis.org

3, 7, 1, 7, 1, 5, 1, 7, 1, 3, 1, 29, 1, 1, 1, 7, 1, 3, 1, 5, 3, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 1, 13, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 17, 1, 1, 1, 7, 1, 3, 1, 7, 3, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 5, 3, 1, 1, 17, 1

Offset: 1

Amiram Eldar, Jun 30 2025

All the terms are odd numbers because if k is even and k*n is not a binary Niven number then so is k*n/2, since A000120(k*n) = A000120(k*n/2).

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

Cf. A000120, A049445, A065878, A144262 (decimal analog), A385482, A385486 (indices of records), A385487 (record values).

a[n_] := Module[{m = n, k = 1}, While[Divisible[m, DigitSum[m, 2]], m += 2*n; k += 2]; k]; Array[a, 100]

a(n) = {my(m = n, k = 1); while(!(m % hammingweight(m)), m += 2*n; k += 2); k;}

from itertools import count
def a(n): return next(k for k in count(1) if (m:=k*n)%m.bit_count() != 0)
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Jun 30 2025

a(n) = 1 if and only if n is in A065878.

cp's OEIS Frontend

A330931 Numbers k such that both k and k + 1 are Niven numbers in base 2 (A049445).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Python

A330932 Starts of runs of 3 consecutive Niven numbers in base 2 (A049445).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

A330933 Starts of runs of 4 consecutive Niven numbers in base 2 (A049445).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

A376616 Binary Niven numbers (A049445) k such that k/wt(k) is also a binary Niven number, where wt(k) = A000120(k) is the binary weight of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A376617 Binary Niven numbers (A049445) k such that m = k/wt(k) and m/wt(m) are also binary Niven numbers, where wt(k) = A000120(k) is the binary weight of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A144302 Odd members of A049445.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A363787 Primitive binary Niven numbers: binary Niven numbers (A049445) that are not twice another binary Niven number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI