A271313 -id:A271313 - 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.

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.

Original entry on oeis.org

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

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 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...

			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.

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

Cf. A080221, A100263, A271313.

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