OFFSET
1,1
COMMENTS
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))
LINKS
Daniel Mondot, Table of n, a(n) for n = 1..20382
EXAMPLE
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.
PROG
(PARI) isok(n) = {nb = 1; for (b=2, n, if ((n % (vecsum(digits(n, b)))) == 0, nb++); ); nb == 7; } \\ Michel Marcus, Apr 03 2016
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Daniel Mondot, Apr 03 2016
STATUS
approved