cp's OEIS Frontend

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.

%I A271311 #19 May 12 2018 02:17:41
%S A271311 6,26,34,122,226,362,514,842,1226,1522,2026,2602,3482,3722,4226,4762,
%T A271311 5042,6242,7226,9026,10202,17162,19322,19882,21026,25282,27226,29242,
%U A271311 30626,32762,38026,39602,40402,42026,43682,47962,48842,53362,60026,68122,73442,75626
%N 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.
%C A271311 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 A271311 They can be separated in 2 distinct groups:
%C A271311 * Most numbers are Harshad numbers in 4 bases that follow pattern A:
%C A271311 - b1 is sqrt(n-1)   (n-1 being a square)
%C A271311 - b2 is n/2
%C A271311 - b3 is n/2 + 1
%C A271311 - b4 is n-1
%C A271311 * Some numbers are Harshad numbers in 4 bases that follow pattern B:
%C A271311 - b1 is 2           (n-1 is not a square)
%C A271311 - b2 is n/2
%C A271311 - b3 is n/2 + 1
%C A271311 - b4 is n-1
%C A271311 This is true for n = 6, 34, 514, 131074, etc...
%H A271311 Daniel Mondot, <a href="/A271311/b271311.txt">Table of n, a(n) for n = 1..103</a>
%e A271311 6 is a Harshad number in bases 2, 3, 4 and 5:              Pattern B
%e A271311 26 is a Harshad number in bases 5, 13, 14 and 25:          Pattern A
%e A271311 34 is a Harshad number in bases 2, 17, 18 and 33:          Pattern B
%e A271311 122 is a Harshad number in bases 11, 61, 62 and 121:       Pattern A
%e A271311 226 is a Harshad number in bases 15, 113, 114 and 225:     Pattern A
%e A271311 362 is a Harshad number in bases 19, 181, 182 and 361:     Pattern A
%e A271311 514 is a Harshad number in bases 2, 257, 258 and 513:      Pattern B
%e A271311 842 is a Harshad number in bases 29, 421, 422 and 841:     Pattern A
%e A271311 1226 is a Harshad number in bases 35, 613, 614 and 1225:   Pattern A
%e A271311 1522 is a Harshad number in bases 39, 761, 762 and 1521:   Pattern A
%e A271311 2026 is a Harshad number in bases 45, 1013, 1014 and 2025: Pattern A
%e A271311 Pattern A: 45=sqrt(2026-1), 1013=2026/2, 1014=2026/2+1, 2025=2026-1
%e A271311 Pattern B: 2=2, 257=514/2, 258=514/2+1, 513=514-1.
%o A271311 (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 A271311 Cf. A080221, A100263, A271313.
%K A271311 nonn,base
%O A271311 1,1
%A A271311 _Daniel Mondot_, Apr 03 2016