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 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...
LINKS
Daniel Mondot, Table of n, a(n) for n = 1..103
EXAMPLE
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.
PROG
(PARI) isok(n) = {nb = 1; for (b=2, n, if ((n % (vecsum(digits(n, b)))) == 0, nb++); ); nb == 6; } \\ Michel Marcus, Apr 03 2016
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Daniel Mondot, Apr 03 2016
STATUS
approved