A330932 -id:A330932 - cp's OEIS frontend

A154701 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Harshad numbers.

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

Offset: 1

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

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

			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.

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

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.

A subset of A005349.

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

#include 
#include 
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;
}

f:=func; 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

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
  fi
od:
Res; # Robert Israel, Feb 01 2019

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 *)

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

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

Original entry on oeis.org

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

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

A331087 Starts of runs of 3 consecutive positive negaFibonacci-Niven numbers (A331085).

Original entry on oeis.org

4, 12, 86, 87, 88, 176, 230, 231, 232, 320, 464, 655, 1194, 1592, 1596, 1854, 1914, 2815, 3016, 3294, 4124, 4178, 4179, 4180, 4268, 4412, 5663, 5755, 8360, 9894, 10614, 10703, 10915, 10975, 13936, 14994, 15114, 15714, 17630, 18976, 19984, 20824, 21835, 23175, 23513

Offset: 1

Amiram Eldar, Jan 08 2020

Numbers of the form F(6*k + 1) - 1, where F(m) is the m-th Fibonacci number, are terms.

Numbers of the form F(k) - 3, where k is congruent to {5, 11, 13, 19} mod 24 (A269819) are starts of runs of 5 consecutive negaFibonacci-Niven numbers.

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

Cf. A000045, A154701, A269819, A328210, A328214, A330932, A331085.

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 = 3; 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

A333428 Starts of runs of 3 consecutive primorial base Niven numbers (A333426).

Original entry on oeis.org

64, 244, 424, 2344, 2524, 4624, 16180, 30064, 30244, 32344, 43900, 60064, 71620, 91408, 99340, 127060, 154780, 182500, 210220, 250936, 338632, 365860, 477280, 510544, 510724, 512824, 513160, 540544, 540880, 790900, 842884, 876988, 1021024, 1021648, 1024000, 1051720

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

			64 is a term since 64, 65 and 66 are all primorial base Niven numbers.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A154701, A328206, A328210, A328214, A330932, A333426, A333427.

max = 7; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; primNivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n, MixedRadix[bases]]]; q1 = primNivenQ[1]; q2 = primNivenQ[2]; seq = {}; Do[q3 = primNivenQ[n]; If[q1 && q2 && q3, AppendTo[seq, n - 2]]; q1 = q2; q2 = q3, {n, 3, nmax}]; seq

A342428 Starts of runs of 3 consecutive Niven numbers in base 3/2 (A342426).

Original entry on oeis.org

2196, 7656, 15624, 16335, 64375, 109224, 171624, 202824, 328887, 329427, 392733, 393640, 447578, 482238, 494450, 520695, 631824, 723519, 773790, 785695, 820960, 876987, 981783, 986607, 1021824, 1026750, 1030455, 1084048, 1108094, 1160670, 1235070, 1242824, 1412908

Offset: 1

Amiram Eldar, Mar 11 2021

			2196 is a term since 2196, 2197 and 2198 are all Niven numbers in base 3/2.

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

Subsequence of A342426 and A342427.
Subsequences: A342429.
Similar sequences: A154701 (decimal), A328206 (factorial), A328210 (Zeckendorf), A328214 (lazy Fibonacci), A330932 (binary), A331087 (negaFibonacci), A333428 (primorial), A334310 (base phi), A331822 (negabinary).

s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[10^6], AllTrue[# + {0, 1, 2}, q] &]

A331822 Starts of runs of 3 consecutive positive negabinary-Niven numbers (A331728).

Original entry on oeis.org

1, 2, 14, 62, 124, 184, 244, 254, 304, 468, 484, 544, 784, 904, 964, 1022, 1084, 1098, 1144, 1264, 1265, 1308, 1448, 1504, 1518, 1924, 1938, 1984, 2044, 2104, 2105, 2358, 2888, 2944, 2945, 3064, 3198, 3248, 3424, 3544, 3604, 3618, 3664, 3828, 3844, 3904, 3964

Offset: 1

Amiram Eldar, Jan 27 2020

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

Cf. A027615, A039724, A154701, A328210, A328214, A330932, A331087, A331090, A331728.

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[n]]; nConsec = 3; neg = negaBinNivenQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec+1; While[c < 50, If[And @@ neg, c++; AppendTo[seq, k - nConsec]]; neg = Join[Rest[neg], {negaBinNivenQ[k]}]; k++]; seq

A334310 Starts of runs of 3 consecutive base phi Niven numbers (A334308).

Original entry on oeis.org

17171, 20760, 29183, 32772, 51336, 65840, 66608, 67990, 89054, 95563, 103682, 108910, 133990, 136512, 167598, 173640, 190094, 197218, 205478, 207364, 223873, 241934, 247115, 248443, 252014, 258816, 261135, 278783, 285129, 285130, 289392, 325934, 326520, 335178

Offset: 1

Amiram Eldar, Apr 22 2020

			17171 is a term since 17171, 17172 and 17173 are all base phi Niven numbers.

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

Cf. A154701, A328206, A328210, A328214, A330932, A333428, A334308, A334309.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n] ]][[1]]; phiNivenQ[n_] := Divisible[n, phiDigSum[n]]; q1 = phiNivenQ[1]; q2 = phiNivenQ[2]; seq = {}; Do[q3 = phiNivenQ[n]; If[q1 && q2 && q3, AppendTo[seq, n - 2]]; q1 = q2; q2 = q3, {n, 3, 300000}]; seq

A344343 Starts of runs of 3 consecutive Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 2, 6, 7, 14, 30, 31, 62, 126, 127, 174, 184, 234, 243, 254, 304, 474, 483, 510, 511, 534, 543, 544, 783, 784, 903, 904, 954, 963, 1022, 1134, 1144, 1253, 1264, 1448, 1475, 1504, 1895, 1914, 1923, 1974, 2046, 2047, 2093, 2094, 2104, 2814, 2888, 2944, 3054, 3064

Offset: 1

Amiram Eldar, May 15 2021

			1 is a term since 1, 2 and 3 are all Gray-code Niven numbers.

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

Cf. A005811, A014550.
Subsequence of A344341 and A344342.
Subsequences: A344344.
Similar sequences: A154701 (decimal), A328206 (factorial), A328210 (Zeckendorf), A328214 (lazy Fibonacci), A330932 (binary), A331087 (negaFibonacci), A333428 (primorial), A334310 (base phi), A331822 (negabinary), A342428 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[3000], AllTrue[# + {0, 1, 2}, gcNivenQ] &]

A351721 Starts of runs of 3 consecutive lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

607068, 640618, 665720, 900921, 1000880, 1375940, 1505878, 1537250, 1924224, 1938508, 1966338, 2527998, 3394224, 3935424, 4242624, 4476624, 4747224, 4794624, 5351367, 5401824, 5526024, 5636356, 5992298, 6103900, 6343298, 7028362, 7113024, 8879424, 8998262, 9431424

Offset: 1

Amiram Eldar, Feb 17 2022

Conjecture: There are no runs of 4 consecutive lazy-Lucas-Niven numbers (checked up to 3*10^9).

			607068 is a term since 607068, 607069 and 607070 are all divisible by the number of terms in their maximal representation:
     k                   A130311(k)  A131343(k)  k/A131343(k)
-------------------------------------------------------------
607068  111010101010101011110111101         18          33726
607069  111010101010101011110111111         19          31951
607070  111010101010101011111010110         17          35710

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000032, A130311, A131343.
Subsequence of A351719 and A351720.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716.

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A154701 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Harshad numbers.

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

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

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A331087 Starts of runs of 3 consecutive positive negaFibonacci-Niven numbers (A331085).

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A333428 Starts of runs of 3 consecutive primorial base Niven numbers (A333426).

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A342428 Starts of runs of 3 consecutive Niven numbers in base 3/2 (A342426).

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A331822 Starts of runs of 3 consecutive positive negabinary-Niven numbers (A331728).

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A334310 Starts of runs of 3 consecutive base phi Niven numbers (A334308).

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