A271311 -id:A271311 - 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.

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

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.

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