A005349 -id:A005349 - cp's OEIS frontend

A330927 Numbers k such that both k and k + 1 are Niven numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 80, 110, 111, 132, 152, 200, 209, 224, 399, 407, 440, 480, 510, 511, 512, 629, 644, 735, 800, 803, 935, 999, 1010, 1011, 1014, 1015, 1016, 1100, 1140, 1160, 1232, 1274, 1304, 1386, 1416, 1455, 1520, 1547, 1651, 1679, 1728, 1853

Offset: 1

Amiram Eldar, Jan 03 2020

Cooper and Kennedy proved that there are infinitely many runs of 20 consecutive Niven numbers. Therefore this sequence is infinite.

			1 is a term since 1 and 1 + 1 = 2 are both Niven numbers.

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

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

Cf. A005349, A060159, A141769, A154701, A328205, A328209, A328213, A330713, A330928, A330929, A330930, A330931.

f:=func; a:=[]; for k in [1..2000] 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

nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; nq1 = nivenQ[1]; seq = {}; Do[nq2 = nivenQ[k]; If[nq1 && nq2, AppendTo[seq, k - 1]]; nq1 = nq2, {k, 2, 2000}]; seq
SequencePosition[Table[If[Divisible[n,Total[IntegerDigits[n]]],1,0],{n,2000}],{1,1}][[;;,1]] (* Harvey P. Dale, Dec 24 2023 *)

from itertools import count, islice
def agen(): # generator of terms
    h1, h2 = 1, 2
    while True:
        if h2 - h1 == 1: yield h1
        h1, h2 = h2, next(k for k in count(h2+1) if k%sum(map(int, str(k))) == 0)
print(list(islice(agen(), 52))) # Michael S. Branicky, Mar 17 2024

A328212 Lazy-Fibonacci-Niven numbers: numbers divisible by the number of terms in their lazy Fibonacci representation (A112310).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 16, 28, 30, 35, 36, 48, 55, 60, 70, 72, 75, 78, 84, 90, 102, 105, 114, 119, 126, 133, 144, 147, 154, 156, 161, 168, 180, 182, 184, 192, 198, 203, 208, 216, 224, 238, 240, 245, 252, 259, 264, 266, 272, 280, 296, 301, 304, 308, 315, 320, 322

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			6 is in the sequence since A112310(6) = 3 and 3 is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A004753, A005349, A112310.

ooQ[n_] := Module[{k = n}, While[k > 3, If[Divisible[k, 4], Return[True], k = Quotient[k, 2]]]; False]; c = 0; s = {}; Do[If[! ooQ[k], c++; d = Total @ IntegerDigits[k, 2]; If[Divisible[c, d], AppendTo[s, c]]], {k, 1, 2000}]; s

A328209 Numbers m such that m and m+1 are consecutive Zeckendorf-Niven numbers (A328208).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 13, 21, 26, 55, 68, 80, 89, 92, 93, 110, 152, 183, 195, 207, 233, 236, 237, 254, 291, 304, 327, 364, 377, 380, 381, 398, 435, 471, 484, 555, 584, 605, 609, 639, 644, 759, 795, 834, 875, 894, 930, 987, 992, 1004, 1011, 1028, 1047, 1076, 1220

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			12 is in the sequence since both 12 and 13 are in A328208: A007895(12) = 3 is a divisor of 12, and A007895(13) = 1 is a divisor of 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A005349, A007895, A328208.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; c = 0; k = 1; s = {}; v = Table[-1, {2}]; While[c < 60, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s (* after Alonso del Arte at A007895 *)

A141769 Beginning of a run of 4 consecutive Niven (or Harshad) numbers.

Original entry on oeis.org

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

Offset: 1

Sergio Pimentel, Sep 15 2008

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

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

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

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.

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).

f:=func; 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

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 *)

{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

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

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

More terms from Amiram Eldar, Jan 03 2020

A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 30, 32, 33, 36, 40, 42, 44, 45, 48, 50, 60, 64, 65, 66, 68, 70, 72, 77, 84, 88, 90, 92, 96, 105, 108, 112, 117, 120, 132, 133, 136, 144, 150, 154, 156, 160, 168, 180, 182, 184, 189, 192, 198, 200, 210, 212, 213, 216, 220

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

Numbers k for which A276086(k) is in A373852. - Antti Karttunen, Jun 22 2024

			1 is a term since A276150(1) = 1 divides 1;
2 is a term since A276150(2) = 1 divides 2;

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A005349, A049345, A049445, A064150, A064438, A064481, A118363, A235168, A276150, A328208, A328212, A373834 (characteristic function)
Cf. A276086, A333427, A333428, A341433, A347496, A358977, A373833, A373852.
Positions of 0's in A373832.

max = 5; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], Divisible[#, sumdig[#]] &]

isA333426 = A373834; \\ Antti Karttunen, Jun 22 2024

A057147 a(n) = n times sum of digits of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 160, 205, 252, 301, 352, 405, 460, 517, 576, 637, 250, 306, 364, 424, 486, 550, 616

Offset: 0

N. J. A. Sloane, Sep 13 2000

A056992(n) = A010888(a(n)). - Reinhard Zumkeller, Mar 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. B. Diniz, About a new family of sequences, arXiv:1607.06082 [math.GM], 2016.

Cf. A007953, A003634, A005349, A052489, A052490, A003635, A245788.

Iterations: A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a057147 n = a007953 n * n  -- Reinhard Zumkeller, Mar 19 2014

for n from 0 to 150 do printf(`%d,`,n*add(convert(n, base, 10)[i], i=1..nops(convert(n,base, 10)))) od:

Table[n*Total[IntegerDigits[n]], {n, 0, 100}]

a(n) = n*sumdigits(n) \\ Franklin T. Adams-Watters, Aug 03 2014

[n*sum([int(d) for d in str(n)]) for n in range(10**5)] # Chai Wah Wu, Aug 05 2014

a(n) = n*A007953(n). - Michel Marcus, Aug 10 2014

G.f.: x * (d/dx) (1/(1 - x))*Sum_{k>=1} (x^k - x^(10^k+k) - 9*x^(10^k))/(1 - x^(10^k)). - Ilya Gutkovskiy, Mar 27 2018

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Sep 13 2000

A328210 Starts of runs of 3 consecutive Zeckendorf-Niven numbers (A328208).

Original entry on oeis.org

1, 2, 3, 4, 12, 92, 236, 380, 1850, 2630, 4184, 7010, 8183, 8360, 11944, 12754, 13550, 16024, 17710, 17714, 18710, 20628, 22323, 22624, 25564, 28910, 31506, 36463, 36484, 39746, 40368, 44694, 48244, 49294, 53543, 58910, 59164, 64743, 70398, 75024, 77874, 78184

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			12 is in the sequence since 12, 13 and 14 are in A328208: A007895(12) = 3 is a divisor of 12, A007895(13) = 1 is a divisor of 13, and A007895(14) = 2 is a divisor of 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Cf. A005349, A007895, A154701, A328208.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; c = 0; k = 1; s = {}; v = Table[-1, {3}]; While[c < 50, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 2]]]; k++]; s (* after Alonso del Arte at A007895 *)

A209871 Quasi-Niven (or Quasi-Harshad) numbers: numbers that divided by the sum of their digits leave 1 as remainder.

Original entry on oeis.org

11, 13, 17, 41, 43, 56, 91, 97, 101, 106, 121, 131, 155, 157, 161, 181, 188, 221, 232, 233, 239, 254, 271, 274, 301, 305, 311, 353, 361, 365, 385, 391, 401, 421, 451, 452, 491, 494, 508, 521, 529, 541, 551, 599, 610, 617, 625, 631, 647, 650, 673, 685, 721

Offset: 1

Paolo P. Lava, Mar 29 2012

Numbers n for which [n mod s(n)]=1, where s(n) is the sum of the digits of n.
z-Niven numbers with A=1 and B=-1 (see comment in A005349).
First pair of consecutive numbers is {232,233}.

			s(43)=7 and 6*7+1=43.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A005349.

[n: n in [1..721] | n mod s eq 1 where s is &+Intseq(n)]; // Bruno Berselli, Mar 29 2012

with(numtheory);
A209871:=proc(i)
local a,b,n;
for n from 1 to i do
  a:=n; b:=0;
  while a>0 do b:=b+(a mod 10); a:=trunc(a/10); od;
  a:=n mod b; if a=1 then print(n); fi;
od; end:
A209871(10000);

A331728 Negabinary-Niven numbers: numbers divisible by the sum of digits in their negabinary representation (A027615).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 33, 35, 36, 40, 42, 48, 50, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 69, 72, 76, 78, 80, 81, 84, 88, 90, 91, 95, 96, 100, 102, 108, 110, 112, 114, 120, 124, 125, 126, 128, 129, 132, 136, 138, 140

Offset: 1

Amiram Eldar, Jan 27 2020

			6 is a term since A039724(6) = 11010 and 1 + 1 + 0 + 1 + 0 = 3 is a divisor of 6.

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

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

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]

A064150 Numbers divisible by the sum of their ternary digits.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 45, 48, 54, 56, 57, 60, 63, 64, 65, 72, 75, 77, 78, 80, 81, 82, 84, 87, 88, 90, 92, 93, 95, 96, 99, 100, 105, 108, 111, 112, 115, 117, 120, 132, 133, 135, 136, 144, 145, 150, 152

Offset: 1

Len Smiley, Sep 11 2001

a(n) mod A053735(a(n)) = 0. - Reinhard Zumkeller, Nov 25 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith).
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166.

Cf. A005349 (Decimal), A049445 (Binary).

a064150 n = a064150_list !! (n-1)
a064150_list = filter (\x -> x `mod` a053735 x == 0) [1..]
-- Reinhard Zumkeller, Oct 28 2012

Select[Range[200], IntegerQ[#/(Plus@@IntegerDigits[#, 3])] &] (* Alonso del Arte, May 27 2011 *)

isok(m)={m % sumdigits(m, 3) == 0} \\ Harry J. Smith, Sep 09 2009

import numpy as np
def gen():
    for dec_num in range(1,153):
        tern_num = np.base_repr(dec_num, 3)
        sum_tern_digits = 0
        for i in tern_num:
            sum_tern_digits += int(i)
        if dec_num % sum_tern_digits == 0:
            yield dec_num
print(list((gen()))) # Adrienne Leonardo, Dec 28 2024

Corrected and extended by Vladeta Jovovic, Sep 22 2001

Offset corrected by Reinhard Zumkeller, Oct 28 2012

cp's OEIS Frontend

A330927 Numbers k such that both k and k + 1 are Niven numbers.

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A328212 Lazy-Fibonacci-Niven numbers: numbers divisible by the number of terms in their lazy Fibonacci representation (A112310).

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A328209 Numbers m such that m and m+1 are consecutive Zeckendorf-Niven numbers (A328208).

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A141769 Beginning of a run of 4 consecutive Niven (or Harshad) numbers.

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A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

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A057147 a(n) = n times sum of digits of n.

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A328210 Starts of runs of 3 consecutive Zeckendorf-Niven numbers (A328208).

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