A271311 - OEIS

%I #19 May 12 2018 02:17:41

%S 6,26,34,122,226,362,514,842,1226,1522,2026,2602,3482,3722,4226,4762,

%T 5042,6242,7226,9026,10202,17162,19322,19882,21026,25282,27226,29242,

%U 30626,32762,38026,39602,40402,42026,43682,47962,48842,53362,60026,68122,73442,75626

%N 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.

%C 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:

%C They can be separated in 2 distinct groups:

%C * Most numbers are Harshad numbers in 4 bases that follow pattern A:

%C - b1 is sqrt(n-1)   (n-1 being a square)

%C - b2 is n/2

%C - b3 is n/2 + 1

%C - b4 is n-1

%C * Some numbers are Harshad numbers in 4 bases that follow pattern B:

%C - b1 is 2           (n-1 is not a square)

%C - b2 is n/2

%C - b3 is n/2 + 1

%C - b4 is n-1

%C This is true for n = 6, 34, 514, 131074, etc...

%H Daniel Mondot, <a href="/A271311/b271311.txt">Table of n, a(n) for n = 1..103</a>

%e 6 is a Harshad number in bases 2, 3, 4 and 5:              Pattern B

%e 26 is a Harshad number in bases 5, 13, 14 and 25:          Pattern A

%e 34 is a Harshad number in bases 2, 17, 18 and 33:          Pattern B

%e 122 is a Harshad number in bases 11, 61, 62 and 121:       Pattern A

%e 226 is a Harshad number in bases 15, 113, 114 and 225:     Pattern A

%e 362 is a Harshad number in bases 19, 181, 182 and 361:     Pattern A

%e 514 is a Harshad number in bases 2, 257, 258 and 513:      Pattern B

%e 842 is a Harshad number in bases 29, 421, 422 and 841:     Pattern A

%e 1226 is a Harshad number in bases 35, 613, 614 and 1225:   Pattern A

%e 1522 is a Harshad number in bases 39, 761, 762 and 1521:   Pattern A

%e 2026 is a Harshad number in bases 45, 1013, 1014 and 2025: Pattern A

%e Pattern A: 45=sqrt(2026-1), 1013=2026/2, 1014=2026/2+1, 2025=2026-1

%e Pattern B: 2=2, 257=514/2, 258=514/2+1, 513=514-1.

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

%Y Cf. A080221, A100263, A271313.

%K nonn,base

%O 1,1

%A _Daniel Mondot_, Apr 03 2016