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.

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

Author

Daniel Mondot, Apr 03 2016

Keywords

• nonn
• base

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...

Examples

			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.
		

Links

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

Crossrefs

Cf. A080221, A100263, A271313.

Programs

PARI

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