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.

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

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 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))

Examples

			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.
		

Links

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

Crossrefs

Cf. A080221, A100263, A271311.

Programs

PARI

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