A330927 -id:A330927

Search: a330927 -id:a330927

Displaying 1-10 of 30 results found.

page 1 2 3

A005349

Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.
(Formerly M0481)

+10
307

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Both spellings, "Harshad" or "harshad", are in use. It is a Sanskrit word, and in Sanskrit there is no distinction between upper- and lower-case letters. - N. J. A. Sloane, Jan 04 2022

z-Niven numbers are numbers n which are divisible by (A*s(n) + B) where A, B are integers and s(n) is sum of digits of n. Niven numbers have A = 1, B = 0. - Ctibor O. Zizka, Feb 23 2008

A070635(a(n)) = 0. A038186 is a subsequence. - Reinhard Zumkeller, Mar 10 2008

A049445 is a subsequence of this sequence. - Ctibor O. Zizka, Sep 06 2010

Complement of A065877; A188641(a(n)) = 1; A070635(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2011

A001101, the Moran numbers, are a subsequence. - Reinhard Zumkeller, Jun 16 2011

A140866 gives the number of terms <= 10^k. - Robert G. Wilson v, Oct 16 2012

The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - Amiram Eldar, Jul 10 2020

From Amiram Eldar, Oct 02 2023: (Start)

Named "Harshad numbers" by the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986) in 1955. The meaning of the word is "giving joy" in Sanskrit.

Named "Niven numbers" by Kennedy et al. (1980) after the Canadian-American mathematician Ivan Morton Niven (1915-1999). During a lecture given at the 5th Annual Miami University Conference on Number Theory in 1977, Niven mentioned a question of finding a number that equals twice the sum of its digits, which appeared in the children's pages of a newspaper. (End)

REFERENCES

Paul Dahlenberg and T. Edgar, Consecutive factorial base Niven numbers, Fib. Q., 56:2 (2018), 163-166.

D. R. Kaprekar, Multidigital Numbers, Scripta Math., Vol. 21 (1955), p. 27.

Robert E. Kennedy and Curtis N. Cooper, On the natural density of the Niven numbers, Abstract 816-11-219, Abstracts Amer. Math. Soc., 6 (1985), 17.

Robert E. Kennedy, Terry A. Goodman, and Clarence H. Best, Mathematical Discovery and Niven Numbers, The MATYC Journal, Vol. 14, No. 1 (1980), pp. 21-25.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 381.

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

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 171.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..11872 (all a(n) <= 100000)

Bob Albrecht, Don Albers, and Jim Conlan, Problem #22 Two-digit Niven numbers, Programming Problems, Recreational Computing Magazine, Vol. 9, No. 1, Issue 46 (1980), p. 59.

Curtis N. Cooper and Robert E. Kennedy, On an asymptotic formula for the Niven numbers, International Journal of Mathematics and Mathematical Sciences, Vol. 8, No. 3 (1985), pp. 537-543.

Curtis N. Cooper and Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly 96 (1989), no. 2, 118-124.

Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.

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.

Nicolas Doyon, Les fascinants nombres de Niven, Thèse de la faculté des sciences et de génie de l'université Laval, Québec,‎ Novembre 2006 (in French).

Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.

Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

Robert E. Kennedy, Digital sums, Niven numbers, and natural density, Crux Mathematicorum, Vol. 8 (1982), pp. 131-135.

Robert E. Kennedy and Curtis N. Cooper, On the natural density of the Niven numbers, The College Mathematics Journal, Vol. 15, No. 4 (Sep., 1984), pp. 309-312.

Project Euler, Harshad Numbers: Problem 387.

Terry Trotter, Niven Numbers for Fun and Profit. [archived page]

Gérard Villemin, Nombres de Harshad (French).

Elaine E. Visitacion, Renalyn T. Boado, Mary Ann V. Doria, and Eduard M. Albay, On Harshad Number, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 134-138. [archived]

Eric Weisstein's World of Mathematics, Digit and Harshad Numbers.

Wikipedia, Harshad number.

EXAMPLE

195 is a term of the sequence because it is divisible by 15 (= 1 + 9 + 5).

MAPLE

s:=proc(n) local N:N:=convert(n, base, 10):sum(N[j], j=1..nops(N)) end:p:=proc(n) if floor(n/s(n))=n/s(n) then n else fi end: seq(p(n), n=1..210); # Emeric Deutsch

MATHEMATICA

harshadQ[n_] := Mod[n, Plus @@ IntegerDigits@ n] == 0; Select[ Range[1000], harshadQ] (* Alonso del Arte, Aug 04 2004 and modified by Robert G. Wilson v, Oct 16 2012 *)

Select[Range[300], Divisible[#, Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Sep 07 2015 *)

PROG

(Haskell)

a005349 n = a005349_list !! (n-1)

a005349_list = filter ((== 0) . a070635) [1..]

-- Reinhard Zumkeller, Aug 17 2011, Apr 07 2011

(Magma) [n: n in [1..250] | n mod &+Intseq(n) eq 0]; // Bruno Berselli, May 28 2011

(Magma) [n: n in [1..250] | IsIntegral(n/&+Intseq(n))]; // Bruno Berselli, Feb 09 2016

(PARI) is(n)=n%sumdigits(n)==0 \\ Charles R Greathouse IV, Oct 16 2012

(Python)

A005349 = [n for n in range(1, 10**6) if not n % sum([int(d) for d in str(n)])] # Chai Wah Wu, Aug 22 2014

(Sage)

[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # Freddy Barrera, Jul 27 2018

(GAP) Filtered([1..230], n-> n mod List(List([1..n], ListOfDigits), Sum)[n]=0); # Muniru A Asiru

CROSSREFS

Cf. A001101, A007602, A007953, A028834, A038186, A049445, A052018, A052019, A052020, A052021, A052022, A065877, A070635, A113315, A188641.

Cf. A001102 (a subsequence).

Cf. A118363 (for factorial-base analog).

Cf. A330927, A154701, A141769, A330928, A330929, A330930 (start of runs of 2, 3, ..., 7 consecutive Niven numbers).

KEYWORD

nonn,base,nice,easy

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

A154701

Numbers k such that k, k + 1 and k + 2 are 3 consecutive Harshad numbers.

+10
33

1, 2, 3, 4, 5, 6, 7, 8, 110, 510, 511, 1010, 1014, 1015, 2022, 2023, 2464, 3030, 3031, 4912, 5054, 5831, 7360, 8203, 9854, 10010, 10094, 10307, 10308, 11645, 12102, 12103, 12255, 12256, 13110, 13111, 13116, 13880, 14704, 15134, 17152, 17575, 18238, 19600, 19682

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Harshad numbers are also known as Niven numbers.

Cooper and Kennedy proved that there are infinitely many runs of 20 consecutive Niven numbers. Therefore this sequence is infinite. - Amiram Eldar, Jan 03 2020

REFERENCES

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.

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.

EXAMPLE

110 is a term since 110 is divisible by 1 + 1 + 0 = 2, 111 is divisible by 1 + 1 + 1 = 3, and 112 is divisible by 1 + 1 + 2 = 4.

MAPLE

Res:= NULL: count:= 0:

state:= 1:

L:= [1]:

for n from 2 while count < 100 do

L[1]:=L[1]+1;

for k from 1 while L[k]=10 do L[k]:= 0;

if k = nops(L) then L:= [0$nops(L), 1]; break

else L[k+1]:= L[k+1]+1 fi

od:

s:= convert(L, `+`);

if n mod s = 0 then

state:= min(state+1, 3);

if state = 3 then count:= count+1; Res:= Res, n-2; fi

else state:= 0

od:

Res; # Robert Israel, Feb 01 2019

MATHEMATICA

nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[3]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 2]], {k, 3, 2*10^4}]; seq (* Amiram Eldar, Jan 03 2020 *)

PROG

#include <conio.h>

int is_harshad(int n){

int i, j, count=0;

i=n;

while(i>0){

count=count+i%10;

i=i/10;

}

return n%count==0?1:0;

}

main(){

int k;

clrscr();

for(k=1; k<=30000; k++)

if(is_harshad(k)&&is_harshad(k+1)&&is_harshad(k+2))

printf("%d, ", k);

getch();

return 0;

}

(Magma) f:=func<n|n mod &+Intseq(n) eq 0>; a:=[]; for k in [1..20000] 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

(Python)

from itertools import count, islice

def agen(): # generator of terms

h1, h2, h3 = 1, 2, 3

while True:

if h3 - h1 == 2: yield h1

h1, h2, h3 = h2, h3, next(k for k in count(h3+1) if k%sum(map(int, str(k))) == 0)

print(list(islice(agen(), 45))) # Michael S. Branicky, Mar 17 2024

CROSSREFS

A subset of A005349.

Cf. A060159, A141769, A328210, A328214, A330927, A330928, A330929, A330930, A330932.

KEYWORD

nonn,base

AUTHOR

Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 14 2009, Jan 15 2009

STATUS

approved

A330931

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

+10
30

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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

LINKS

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.

EXAMPLE

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

MATHEMATICA

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

PROG

(Magma) f:=func<n|n mod &+Intseq(n, 2) eq 0>; 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

(Python)

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

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Amiram Eldar, Jan 03 2020

STATUS

approved

A141769

Beginning of a run of 4 consecutive Niven (or Harshad) numbers.

+10
26

1, 2, 3, 4, 5, 6, 7, 510, 1014, 2022, 3030, 10307, 12102, 12255, 13110, 60398, 61215, 93040, 100302, 101310, 110175, 122415, 127533, 131052, 131053, 196447, 201102, 202110, 220335, 223167, 245725, 255045, 280824, 306015, 311232, 318800, 325600, 372112, 455422

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Cooper and Kennedy proved that there are infinitely many runs of 20 consecutive Niven numbers. Therefore this sequence is infinite. - Amiram Eldar, Jan 03 2020

REFERENCES

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.

LINKS

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

Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.

Helen G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.

Eric Weisstein's World of Mathematics, Harshad Number.

Wikipedia, Harshad number.

Brad Wilson Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.

FORMULA

This A141769 = { A005349(k) | A005349(k+3) = A005349(k)+3 }. - M. F. Hasler, Jan 03 2022

EXAMPLE

510 is in the sequence because 510, 511, 512 and 513 are all Niven numbers.

MATHEMATICA

nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[4]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 3]], {k, 4, 5*10^5}]; seq (* Amiram Eldar, Jan 03 2020 *)

PROG

(Magma) f:=func<n|n mod &+Intseq(n) eq 0>; a:=[]; for k in [1..500000] 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

(PARI) {A141769_first( N=50, L=4, a=List())= for(n=1, oo, n+=L; for(m=1, L, n--%sumdigits(n) && next(2)); listput(a, n); N--|| break); a} \\ M. F. Hasler, Jan 03 2022

(Python)

from itertools import count, islice

def agen(): # generator of terms

h1, h2, h3, h4 = 1, 2, 3, 4

while True:

if h4 - h1 == 3: yield h1

h1, h2, h3, h4, = h2, h3, h4, next(k for k in count(h4+1) if k%sum(map(int, str(k))) == 0)

print(list(islice(agen(), 40))) # Michael S. Branicky, Mar 17 2024

CROSSREFS

Cf. A005349, A330927, A154701, A330928, A330929, A330930, A060159 (start of run of 1, 2, ..., 7, exactly n consecutive Harshad numbers).

Cf. A330933, A328211, A328215 (analog for base 2, Zeckendorf- resp. Fibonacci-Niven variants).

KEYWORD

base,nonn

AUTHOR

Sergio Pimentel, Sep 15 2008

EXTENSIONS

More terms from Amiram Eldar, Jan 03 2020

STATUS

approved

A331086

Positive numbers k such that k and k + 1 are both negaFibonacci-Niven numbers (A331085).

+10
18

1, 4, 5, 9, 12, 13, 26, 68, 86, 87, 88, 89, 93, 99, 155, 176, 177, 183, 195, 212, 230, 231, 232, 233, 237, 243, 255, 320, 321, 327, 384, 395, 411, 415, 424, 464, 465, 471, 475, 484, 515, 544, 575, 591, 602, 644, 655, 656, 744, 824, 875, 894, 924, 1043, 1115, 1127

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Fibonacci numbers F(6*k - 1) and F(6*k) are terms.

LINKS

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

MATHEMATICA

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]];

f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i];

negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s];

negFibQ[n_] := Divisible[n, negaFibTermsNum[n]];

nConsec = 2; neg = negFibQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec + 1; While[c < 55, If[And @@ neg, c++; AppendTo[seq, k - nConsec]]; neg = Join[Rest[neg], {negFibQ[k]}]; k++]; seq

CROSSREFS

Cf. A328209, A328213, A330927, A330931, A331085.

KEYWORD

nonn,base

AUTHOR

Amiram Eldar, Jan 08 2020

STATUS

approved

A333427

Numbers k such that k and k+1 are both primorial base Niven numbers (A333426).

+10
18

1, 8, 24, 32, 44, 64, 65, 132, 212, 224, 244, 245, 296, 368, 424, 425, 468, 560, 656, 720, 728, 737, 869, 1056, 1088, 1416, 1572, 1728, 2100, 2312, 2324, 2344, 2345, 2524, 2525, 2568, 2600, 2672, 2820, 2960, 3032, 3132, 3156, 3200, 3288, 3392, 3444, 4096, 4424

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

1 is a term since 1 and 2 are both primorial base Niven numbers.

MATHEMATICA

max = 6; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; primNivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n, MixedRadix[bases]]]; q1 = primNivenQ[1]; seq = {}; Do[q2 = primNivenQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, nmax}]; seq

CROSSREFS

Cf. A328205, A328209, A328213, A330927, A330931, A333426.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Mar 20 2020

STATUS

approved

A331820

Positive numbers k such that k and k + 1 are both negabinary-Niven numbers (A331728).

+10
16

1, 2, 3, 8, 14, 15, 20, 32, 35, 56, 62, 63, 68, 80, 90, 95, 124, 125, 128, 174, 184, 185, 215, 224, 244, 245, 248, 254, 255, 260, 272, 275, 300, 304, 305, 320, 335, 342, 468, 469, 484, 485, 512, 515, 544, 545, 552, 575, 594, 636, 720, 762, 784, 785, 804, 846, 896

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

8 is a term since both 8 and 8 + 1 = 9 are negabinary-Niven numbers: A039724(8) = 11000 and 1 + 1 + 0 + 0 + 0 = 2 is a divisor of 8, and A039724(9) = 11001 and 1 + 1 + 0 + 0 + 1 = 3 is a divisor of 9.

MATHEMATICA

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[negaBinNivenQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s

CROSSREFS

Cf. A027615, A039724, A328209, A328213, A330927, A330931, A331086, A331089, A331728.

KEYWORD

nonn,base

AUTHOR

Amiram Eldar, Jan 27 2020

STATUS

approved

A334309

Numbers k such that k and k+1 are both base phi Niven numbers (A334308).

+10
16

1, 15, 35, 90, 95, 231, 644, 728, 944, 1016, 1110, 1331, 1629, 1736, 1770, 1899, 1925, 2232, 2255, 2384, 2456, 2629, 2652, 2760, 3104, 3176, 3288, 3444, 3729, 3789, 3860, 4410, 4415, 4509, 4544, 4718, 4939, 4960, 5229, 5239, 5489, 5789, 5831, 5984, 6039, 6111

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

1 is a term since 1 and 2 are both base phi Niven numbers.

MATHEMATICA

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; Select[Range[6000], phiNivenQ[#] && phiNivenQ[# + 1] &]

CROSSREFS

Cf. A328205, A328209, A328213, A330927, A330931, A333427, A334308.

KEYWORD

nonn,base

AUTHOR

Amiram Eldar, Apr 22 2020

STATUS

approved

A342427

Numbers k such that k and k + 1 are both Niven numbers in base 3/2 (A342426).

+10
16

1, 168, 459, 1817, 2196, 2197, 2655, 3128, 3280, 3699, 4199, 4575, 4927, 5184, 5795, 6600, 7215, 7259, 7656, 7657, 8448, 9636, 11304, 11339, 12492, 14160, 14175, 14424, 14805, 15624, 15625, 16335, 16336, 16925, 17802, 19170, 20349, 20811, 21624, 21735, 22197

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

168 is a term since both 168 and 169 are Niven numbers in base 3/2. 168 in base 3/2 is 2120220210 and 2+1+2+0+2+2+0+2+1+0 = 12 is a divisor of 168. 169 in base 3/2 is 2120220211 and 2+1+2+0+2+2+0+2+1+1 = 13 is a divisor of 169.

MATHEMATICA

s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[22000], q[#] && q[# + 1] &]

CROSSREFS

Subsequence of A342426.

Subsequences: A342428 and A342429.

Similar sequences: A330927 (decimal), A328205 (factorial), A328209 (Zeckendorf), A328213 (lazy Fibonacci), A330931 (binary), A331086 (negaFibonacci), A333427 (primorial), A334309 (base phi), A331820 (negabinary).

KEYWORD

nonn,base

AUTHOR

Amiram Eldar, Mar 11 2021

STATUS

approved

A344342

Numbers k such that k and k + 1 are both Gray-code Niven numbers (A344341).

+10
15

1, 2, 3, 6, 7, 8, 14, 15, 27, 30, 31, 32, 39, 44, 51, 56, 62, 63, 75, 99, 104, 111, 123, 126, 127, 128, 135, 144, 155, 159, 174, 175, 184, 185, 195, 204, 207, 215, 224, 231, 234, 235, 243, 244, 248, 254, 255, 264, 275, 284, 294, 300, 304, 305, 315, 335, 354, 375

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

1 is a term since 1 and 2 are both Gray-code Niven numbers.

MATHEMATICA

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[400], And @@ gcNivenQ[# + {0, 1}] &]

CROSSREFS

Cf. A005811, A014550.

Subsequence of: A344341.

Subsequences: A344343 and A344344.

KEYWORD

nonn,base

AUTHOR

Amiram Eldar, May 15 2021

STATUS

approved

page 1 2 3

Search completed in 0.020 seconds