A080221 -id:A080221 - cp's OEIS frontend

A100263 Values of n such that A080221(n)=5; i.e., values of n such that n is divisible by the sum of digits of n when expressed in exactly 5 of the bases b=1...n.

9, 14, 22, 38, 46, 58, 62, 74, 86, 94, 106, 118, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 526, 538, 542, 554, 562, 566, 586, 614

Offset: 1

John W. Layman, Nov 10 2004

It appears that, except for the first term a(1)=9, each term of this sequence is twice a prime.

Besides base 1, and bases b>=n (bases greater than or equal to the number itself), for which any number can be a Harshad number, these numbers are Harshad numbers in 3 other bases (where b=2..n-1): b1, b2, and b3, where b1 is n/2, b2 is n/2 + 1, b3 is n-1. Except for a(1)=9 that is a Harshad number in bases 3, 4 and 7. - Daniel Mondot, Apr 03 2016

			9 is a Harshad number in bases 3, 4 and 7 (not following pattern);
14 is a Harshad number in bases  7,  8 and 13;
22 is a Harshad number in bases 11, 12 and 21;
38 is a Harshad number in bases 19, 20 and 37;
46 is a Harshad number in bases 23, 24 and 45;
58 is a Harshad number in bases 29, 30 and 57;
62 is a Harshad number in bases 31, 32 and 61;
74 is a Harshad number in bases 37, 38 and 73;
86 is a Harshad number in bases 43, 44 and 85;
94 is a Harshad number in bases 47, 48 and 93;
47 = 94/2, 48 = 94/2 + 1, 93 = 94 - 1. - _Daniel Mondot_, Apr 03 2016

Daniel Mondot, Table of n, a(n) for n = 1..29865

Cf. A080221, A271311, A271313.

A271311 Values of n such that A080221(n)=6; i.e., values of n such that n is divisible by the sum of digits of n when expressed in exactly 6 of the bases b=1...n.

Original entry on oeis.org

6, 26, 34, 122, 226, 362, 514, 842, 1226, 1522, 2026, 2602, 3482, 3722, 4226, 4762, 5042, 6242, 7226, 9026, 10202, 17162, 19322, 19882, 21026, 25282, 27226, 29242, 30626, 32762, 38026, 39602, 40402, 42026, 43682, 47962, 48842, 53362, 60026, 68122, 73442, 75626

Offset: 1

Daniel Mondot, Apr 03 2016

Besides base 1, and bases b>=n (bases greater than or equal to the number itself), for which any number can be a Harshad number, these numbers are Harshad numbers in 4 other bases (where b=2...n-1): b1, b2, b3, and b4, where:
They can be separated in 2 distinct groups:
* Most numbers are Harshad numbers in 4 bases that follow pattern A:
- b1 is sqrt(n-1)   (n-1 being a square)
- b2 is n/2
- b3 is n/2 + 1
- b4 is n-1
* Some numbers are Harshad numbers in 4 bases that follow pattern B:
- b1 is 2           (n-1 is not a square)
- b2 is n/2
- b3 is n/2 + 1
- b4 is n-1
This is true for n = 6, 34, 514, 131074, etc...

			6 is a Harshad number in bases 2, 3, 4 and 5:              Pattern B
26 is a Harshad number in bases 5, 13, 14 and 25:          Pattern A
34 is a Harshad number in bases 2, 17, 18 and 33:          Pattern B
122 is a Harshad number in bases 11, 61, 62 and 121:       Pattern A
226 is a Harshad number in bases 15, 113, 114 and 225:     Pattern A
362 is a Harshad number in bases 19, 181, 182 and 361:     Pattern A
514 is a Harshad number in bases 2, 257, 258 and 513:      Pattern B
842 is a Harshad number in bases 29, 421, 422 and 841:     Pattern A
1226 is a Harshad number in bases 35, 613, 614 and 1225:   Pattern A
1522 is a Harshad number in bases 39, 761, 762 and 1521:   Pattern A
2026 is a Harshad number in bases 45, 1013, 1014 and 2025: Pattern A
Pattern A: 45=sqrt(2026-1), 1013=2026/2, 1014=2026/2+1, 2025=2026-1
Pattern B: 2=2, 257=514/2, 258=514/2+1, 513=514-1.

Daniel Mondot, Table of n, a(n) for n = 1..103

Cf. A080221, A100263, A271313.

isok(n) = {nb = 1; for (b=2, n, if ((n % (vecsum(digits(n, b)))) == 0, nb++);); nb == 6;} \\ Michel Marcus, Apr 03 2016

A271313 Values of n such that A080221(n)=7; i.e., values of n such that n is divisible by the sum of digits of n when expressed in exactly 7 of the bases b=1...n.

Original entry on oeis.org

8, 10, 25, 49, 82, 121, 141, 159, 177, 213, 219, 237, 267, 303, 309, 411, 417, 447, 453, 471, 501, 519, 529, 537, 543, 573, 591, 597, 626, 633, 669, 681, 699, 717, 753, 771, 789, 807, 831, 849, 879, 921, 933, 939, 951, 1047, 1077, 1119, 1137, 1149, 1167, 1203

Offset: 1

Daniel Mondot, Apr 03 2016

Besides base 1, and bases b>=n (bases greater than or equal to the number itself), for which any number can be a Harshad number, these numbers are Harshad numbers in 5 other bases (where b=2...n-1): b1, b2, b3, b4, and b5, where:
They can be separated in 4 distinct groups.
* The first 3 entries (n=8, 10 and 25) are Harshad numbers in bases that do not follow other patterns.
* Most numbers are Harshad numbers in 5 bases that follow pattern A:
  - b1 = n/3
  - b2 = n/3+1
  - b3 = (n-1)/2
  - b4 = 2*n/3+1
  - b5 = n-2
* Some numbers are Harshad numbers in 5 bases that follow pattern B:
  - b1 = sqrt(sqrt(n-1))
  - b2 = sqrt(n-1)
  - b3 = n/2
  - b4 = (n/2)+1
  - b5 = n-1
* Some numbers are Harshad numbers in 5 bases that follow pattern C:
  - b1 = sqrt(n)
  - b2 = sqrt(n)+1
  - b3 = 2*sqrt(n)+1
  - b4 = (n+2-sqrt(n))/2
  - b5 = (n+1-sqrt(n))

			8 is a Harshad number in bases 2, 3, 4, 5 and 7:            no pattern
10 is a Harshad number in bases 2, 3, 5, 6 and 9:           no pattern
25 is a Harshad number in bases 3, 5, 6, 11 and 21:         no pattern
49 is a Harshad number in bases 7, 8, 15, 22 and 43:        pattern C
82 is a Harshad number in bases 3, 9, 41, 42 and 81:        pattern B
121 is a Harshad number in bases 11, 12, 23, 56 and 111:    pattern C
141 is a Harshad number in bases 47, 48, 70, 95 and 139:    pattern A
159 is a Harshad number in bases 53, 54, 79, 107 and 157:   pattern A
177 is a Harshad number in bases 59, 60, 88, 119 and 175:   pattern A
213 is a Harshad number in bases 71, 72, 106, 143 and 211:  pattern A
219 is a Harshad number in bases 73, 74, 109, 147 and 217:  pattern A
237 is a Harshad number in bases 79, 80, 118, 159 and 235:  pattern A
267 is a Harshad number in bases 89, 90, 133, 179 and 265:  pattern A
Pattern A: 47=141/3, 48=141/3+1, 70=(141-1)/2, 95=(2*141/3)+1, 139=141-2
Pattern B: 3=sqrt(sqrt(82-1)), 9=sqrt(82-1), 41=82/2, 42=82/2+1, 81=82-1
Pattern C: 7=sqrt(49), 8=sqrt(49)+1, 15=2*sqrt(49)+1, 22=(49+2-sqrt(49))/2, 43=49+1-sqrt(49)
List of n that follow pattern B: 82, 626, 2402, 14642, 28562, 83522, etc...
List of n that follow pattern C: 49, 121, 529, 2209, 3481, 6889, 11449, 27889, 32041, 51529, 69169, 120409, 128881, etc...
List of n that follow pattern A: all others not already mentioned above.

Daniel Mondot, Table of n, a(n) for n = 1..20382

Cf. A080221, A100263, A271311.

isok(n) = {nb = 1; for (b=2, n, if ((n % (vecsum(digits(n, b)))) == 0, nb++);); nb == 7;} \\ Michel Marcus, Apr 03 2016

A356555 Irregular triangle T(n, k), n > 0, k = 1..A080221(n) read by rows; the n-th row contains, in ascending order, the bases b from 2..n+1 where the sum of digits of n divides n.

Original entry on oeis.org

2, 2, 3, 3, 4, 2, 3, 4, 5, 5, 6, 2, 3, 4, 5, 6, 7, 7, 8, 2, 3, 4, 5, 7, 8, 9, 3, 4, 7, 9, 10, 2, 3, 5, 6, 9, 10, 11, 11, 12, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 13, 14, 7, 8, 13, 14, 15, 3, 5, 6, 7, 11, 13, 15, 16, 2, 3, 4, 5, 7, 8, 9, 13, 15, 16, 17, 17, 18

Offset: 1

Rémy Sigrist, Aug 12 2022

A080221 provides row lengths (note that for n > 0, we consider the base n+1 but not the base 1, unlike A080221 that considers the base 1 but not the base n+1, however this does not matter as the sums of digits of n in base 1 and base n+1 are the same).

			Triangle T(n, k) begins:
    n    n-th row
    --   --------
     1   [2]
     2   [2, 3]
     3   [3, 4]
     4   [2, 3, 4, 5]
     5   [5, 6]
     6   [2, 3, 4, 5, 6, 7]
     7   [7, 8]
     8   [2, 3, 4, 5, 7, 8, 9]
     9   [3, 4, 7, 9, 10]
    10   [2, 3, 5, 6, 9, 10, 11]
    11   [11, 12]
    12   [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13]
    13   [13, 14]
    14   [7, 8, 13, 14, 15]
    15   [3, 5, 6, 7, 11, 13, 15, 16]
    16   [2, 3, 4, 5, 7, 8, 9, 13, 15, 16, 17]
    17   [17, 18]

Cf. A080221, A356552.

row(n) = select(b -> n % sumdigits(n,b)==0, [2..n+1])

from sympy.ntheory import digits
def row(n): return [b for b in range(2, n+2) if n%sum(digits(n, b)[1:])==0]
print([an for n in range(1, 18) for an in row(n)]) # Michael S. Branicky, Aug 12 2022

T(n, 1) = A356552(n).
T(n, A080221(n)-1) = n for n > 1.
T(n, A080221(n)) = n+1.

A330813 Numbers k that are Niven numbers in a record number of bases 1 <= b <= k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 24, 36, 48, 60, 72, 96, 120, 144, 168, 180, 240, 336, 360, 480, 600, 630, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 83160

Offset: 1

Amiram Eldar, Jan 01 2020

Indices of records of A080221.

			4 is a term since it is a Niven number in 4 bases: 1, 2, 3, 4, while the numbers below 4 are Niven numbers in less than 4 bases.

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A005349, A049445, A080221.

nivenQ[n_, b_] := Divisible[n, Total @ IntegerDigits[n, b]]; basesCount[n_] := 1 + Sum[Boole @ nivenQ[n, b], {b, 2, n}]; bmax = 0; seq = {}; Do[b = basesCount[n]; If[b > bmax ,bmax = b; AppendTo[seq,n]],{n,1,1000}];seq

A341434 a(n) is the number of bases 1 < b < n in which n is divisible by its product of digits.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 2, 2, 1, 5, 2, 3, 4, 6, 1, 5, 1, 5, 4, 4, 1, 9, 2, 2, 4, 5, 1, 7, 3, 9, 4, 2, 3, 12, 1, 2, 3, 10, 1, 7, 2, 7, 7, 2, 1, 15, 2, 5, 3, 6, 1, 10, 3, 10, 4, 3, 1, 14, 1, 2, 7, 14, 3, 8, 1, 6, 3, 6, 1, 20, 2, 3, 8, 7, 3, 7, 1, 16, 7, 2, 1, 14

Offset: 1

Amiram Eldar, Feb 11 2021

			a(3) = 1 since 3 is divisible by its product of digits only in base 2: 3 = 11_2 and 1*1 | 3.
a(6) = 2 since 6 is divisible by its product of digits in 2 bases: in base 4, 6 = 12_4 and 1*2 | 6, and in base 5, 6 = 11_5 and 1*1 | 6.

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

Cf. A007602, A068953, A080221, A085104, A088323, A119598.

q[n_, b_] := (p = Times @@ IntegerDigits[n, b]) > 0 && Divisible[n, p]; a[n_] := Count[Range[2, n], _?(q[n, #] &)]; Array[a, 100]

a(n) = sum(b=2, n-1, my(x=vecprod(digits(n, b))); x && !(n%x)); \\ Michel Marcus, Feb 12 2021

a(n) > 0 for all numbers n > 2 since n in base b = n-1 is 11.
a(n) > 1 for all even numbers > 4 since n in base b = n-2 is 12. Similarly, a(n) > 1 for all composite numbers > 4 since if n = k*m, then n is divisible by its product of digits in bases n-m and n-k.
a(p) > 1 for primes p in A085104.
a(p) > 2 for primes p in A119598 (i.e., 31, 8191, ...).
a(n) >= A088323(n), with equality if n = 4 or if n is a prime.

A357823 a(n) is the number of bases > 1 where n is not divisible by the sum of its digits.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 5, 1, 4, 3, 9, 1, 11, 9, 7, 5, 15, 5, 17, 7, 11, 17, 21, 5, 18, 20, 17, 14, 27, 12, 29, 16, 24, 28, 24, 13, 35, 33, 31, 17, 39, 22, 41, 33, 26, 41, 45, 18, 42, 34, 42, 38, 51, 33, 45, 35, 48, 53, 57, 26, 59, 57, 44, 41, 52, 43, 65, 56, 60, 48

Offset: 1

Rémy Sigrist, Oct 17 2022

The sequence is well defined as the sum of digits of n equals n (and hence divides n) in any base > n.

			For n = 10, we have:
       b  sum of digits  divisible?
    ----  -------------  ----------
       2              2  Yes
       3              2  Yes
       4              4  No
       5              2  Yes
       6              5  Yes
       7              4  No
       8              3  No
       9              2  Yes
      10              1  Yes
    >=11             10  Yes
so a(n) = #{ 4, 7, 8 } = 3.

Cf. A080221, A138530, A356555.

NivenQ[n_, b_] := Divisible[n, Total @ IntegerDigits[n, b]]; a[n_] := Sum[Boole @ !NivenQ[n, b], {b, 2, n}]; Array[a, 70]

a(n) = sum(b=2, n, n%sumdigits(n,b)!=0)

from sympy.ntheory.factor_ import digits
def A357823(n): return sum(1 for b in range(2,n) if n%sum(digits(n,b)[1:])) # Chai Wah Wu, Oct 19 2022

a(n) = n - A080221(n).

a(p) = p - 2 for any prime number p.

A330812 Least number >= n that is a Niven number in all bases 1 <= b <= n.

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 12, 24, 24, 24, 24, 24, 24, 432, 720, 720, 720, 720, 720, 840, 840, 840, 3360, 13860, 13860, 13860, 13860, 13860, 40320, 100800, 100800, 2106720, 7698600, 9028800, 9028800, 9028800, 9028800, 9028800, 9028800, 9028800, 9028800, 9028800, 9028800

Offset: 1

Amiram Eldar, Jan 01 2020

			a(4) = 4 since the representations of 4 in bases 1 to 4 are 1111, 100, 11, 10, the corresponding sums of digits are 4, 1, 2, and 1, and all are divisors of 4. Thus 4 is a Niven number in bases 1, 2, 3, and 4, and it is the least number with this property.

Cf. A005349, A049445, A080221, A226169.

A[1]:= 1: m:= 1:
for n from 2 while m < 30 do
   kk:= n;
   for k from 2 to n-1 do
     if n mod convert(convert(n,base,k),`+`) <> 0 then kk:= k-1; break fi;
     od;
   if kk > m then
     for k from m+1 to kk do A[k]:= n od;
     m:= kk;
   fi
od:
seq(A[k],k=1..m); # Robert Israel, Jan 01 2020

nivenQ[n_, b_] := Divisible[n, Total @ IntegerDigits[n,b]]; a[n_] := Module[{k = n}, While[!AllTrue[Range[2, n], nivenQ[k, #] &], k++]; k]; Array[a,30]

cp's OEIS Frontend

A100263 Values of n such that A080221(n)=5; i.e., values of n such that n is divisible by the sum of digits of n when expressed in exactly 5 of the bases b=1...n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A271311 Values of n such that A080221(n)=6; i.e., values of n such that n is divisible by the sum of digits of n when expressed in exactly 6 of the bases b=1...n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A271313 Values of n such that A080221(n)=7; i.e., values of n such that n is divisible by the sum of digits of n when expressed in exactly 7 of the bases b=1...n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A356555 Irregular triangle T(n, k), n > 0, k = 1..A080221(n) read by rows; the n-th row contains, in ascending order, the bases b from 2..n+1 where the sum of digits of n divides n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Formula

A330813 Numbers k that are Niven numbers in a record number of bases 1 <= b <= k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A341434 a(n) is the number of bases 1 < b < n in which n is divisible by its product of digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A357823 a(n) is the number of bases > 1 where n is not divisible by the sum of its digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A330812 Least number >= n that is a Niven number in all bases 1 <= b <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs