cp's OEIS Frontend

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.

%I A271313 #18 May 12 2018 05:24:20
%S A271313 8,10,25,49,82,121,141,159,177,213,219,237,267,303,309,411,417,447,
%T A271313 453,471,501,519,529,537,543,573,591,597,626,633,669,681,699,717,753,
%U A271313 771,789,807,831,849,879,921,933,939,951,1047,1077,1119,1137,1149,1167,1203
%N 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.
%C A271313 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:
%C A271313 They can be separated in 4 distinct groups.
%C A271313 * The first 3 entries (n=8, 10 and 25) are Harshad numbers in bases that do not follow other patterns.
%C A271313 * Most numbers are Harshad numbers in 5 bases that follow pattern A:
%C A271313   - b1 = n/3
%C A271313   - b2 = n/3+1
%C A271313   - b3 = (n-1)/2
%C A271313   - b4 = 2*n/3+1
%C A271313   - b5 = n-2
%C A271313 * Some numbers are Harshad numbers in 5 bases that follow pattern B:
%C A271313   - b1 = sqrt(sqrt(n-1))
%C A271313   - b2 = sqrt(n-1)
%C A271313   - b3 = n/2
%C A271313   - b4 = (n/2)+1
%C A271313   - b5 = n-1
%C A271313 * Some numbers are Harshad numbers in 5 bases that follow pattern C:
%C A271313   - b1 = sqrt(n)
%C A271313   - b2 = sqrt(n)+1
%C A271313   - b3 = 2*sqrt(n)+1
%C A271313   - b4 = (n+2-sqrt(n))/2
%C A271313   - b5 = (n+1-sqrt(n))
%H A271313 Daniel Mondot, <a href="/A271313/b271313.txt">Table of n, a(n) for n = 1..20382</a>
%e A271313 8 is a Harshad number in bases 2, 3, 4, 5 and 7:            no pattern
%e A271313 10 is a Harshad number in bases 2, 3, 5, 6 and 9:           no pattern
%e A271313 25 is a Harshad number in bases 3, 5, 6, 11 and 21:         no pattern
%e A271313 49 is a Harshad number in bases 7, 8, 15, 22 and 43:        pattern C
%e A271313 82 is a Harshad number in bases 3, 9, 41, 42 and 81:        pattern B
%e A271313 121 is a Harshad number in bases 11, 12, 23, 56 and 111:    pattern C
%e A271313 141 is a Harshad number in bases 47, 48, 70, 95 and 139:    pattern A
%e A271313 159 is a Harshad number in bases 53, 54, 79, 107 and 157:   pattern A
%e A271313 177 is a Harshad number in bases 59, 60, 88, 119 and 175:   pattern A
%e A271313 213 is a Harshad number in bases 71, 72, 106, 143 and 211:  pattern A
%e A271313 219 is a Harshad number in bases 73, 74, 109, 147 and 217:  pattern A
%e A271313 237 is a Harshad number in bases 79, 80, 118, 159 and 235:  pattern A
%e A271313 267 is a Harshad number in bases 89, 90, 133, 179 and 265:  pattern A
%e A271313 Pattern A: 47=141/3, 48=141/3+1, 70=(141-1)/2, 95=(2*141/3)+1, 139=141-2
%e A271313 Pattern B: 3=sqrt(sqrt(82-1)), 9=sqrt(82-1), 41=82/2, 42=82/2+1, 81=82-1
%e A271313 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)
%e A271313 List of n that follow pattern B: 82, 626, 2402, 14642, 28562, 83522, etc...
%e A271313 List of n that follow pattern C: 49, 121, 529, 2209, 3481, 6889, 11449, 27889, 32041, 51529, 69169, 120409, 128881, etc...
%e A271313 List of n that follow pattern A: all others not already mentioned above.
%o A271313 (PARI) isok(n) = {nb = 1; for (b=2, n, if ((n % (vecsum(digits(n, b)))) == 0, nb++);); nb == 7;} \\ _Michel Marcus_, Apr 03 2016
%Y A271313 Cf. A080221, A100263, A271311.
%K A271313 nonn,base
%O A271313 1,1
%A A271313 _Daniel Mondot_, Apr 03 2016