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Search: id:A080221
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| A080221 |
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n is Harshad (divisible by the sum of its digits) in a(n) bases from 1 to n. |
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+0
2
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| 1, 2, 2, 4, 2, 6, 2, 7, 5, 7, 2, 11, 2, 5, 8, 11, 2, 13, 2, 13, 10, 5, 2, 19, 7, 6, 10, 14, 2, 18, 2, 16, 9, 6, 11, 23, 2, 5, 8, 23, 2, 20, 2, 11, 19, 5, 2, 30, 7, 16, 9, 14, 2, 21, 10, 21, 9, 5, 2, 34, 2, 5, 19, 23, 13, 23, 2, 12, 9, 22, 2, 39, 2, 5, 20, 13, 13, 21, 2, 34, 18, 7, 2, 37, 12, 5
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OFFSET
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1,2
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COMMENT
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For non-composite integers, a(n)=d(n) (cf. A000005); for composite integers, a(n)> d(n). a(n) < n for all n > 6.
It appears that a(n) never takes on the value 3. Is there a proof of this? See A100263 for the sequence of values of n for which a(n)=5. It appears that, except for n=9, all values of n such that a(n) is 5 or 6 are twice a prime. - John W. Layman (layman(AT)math.vt.edu), Nov 10 2004
a(n) is never 3. As noted, 1 or any prime has a(n) = d(n) < 3. The only composites with d(n) <= 3 are squares of primes, for which d(n) = 3. But p^2 has the representation (p-1)(1) in base (p+1), so a(p^2) >= 4. Any product of two distinct odd numbers n = ab with 1<a<b can be written as a,0 in base b; 1,(a-1) in base ab-a+1; 1,(b-1) in base ab-b+1; 2,a-2 in base a(b-1)/2+1; and 2,b-2 in base (a-1)b/2+1; plus 1 and n work for any n, so a(n)>6. If n = a^2, with a>3, we have 1,0 in base a; (a-1)1 in base a+1; 1,(a-1) in base a^2-a+1; 2,(a-2) in base a(a-1)/2+1; and (a-1)/2,(a+1)/2 in base 2a+1; together with 1 and n this means a(n)>6 for this form, too. Similar considerations eliminate other forms, leaving only 2p as possible values to have a(n) = 5 or 6. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 03 2006
It is easy to prove that only 1, 2, 4 and 6 are all-Harshad numbers (numbers which are divisible by the sum of their digits in every base). - Adam Kertesz (Kertesz.Adam(AT)gmail.com), Feb 04 2008
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REFERENCES
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Eric W. Weisstein: CRC Concise Encyclopedia of Mathematics, Second ed.,Chapman & Hall/CRC, 2003, p. 1310
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EXAMPLE
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6 is represented by the numeral 111111 in unary, 110 in binary, 20 in base 3, 12 in base 4, 11 in base 5, and 10 in base 6. The sums of the digits are 6, 2, 2, 3, 2, and 1 respectively, all divisors of 6; therefore a(6)=6.
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CROSSREFS
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See A005349 for numbers that are Harshad in base 10.
Cf. A100263.
Adjacent sequences: A080218 A080219 A080220 this_sequence A080222 A080223 A080224
Sequence in context: A128982 A096216 A121599 this_sequence A137849 A118982 A129457
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KEYWORD
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nonn
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AUTHOR
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Matthew Vandermast (ghodges14(AT)comcast.net), Mar 16 2003
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EXTENSIONS
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More terms from John W. Layman (layman(AT)math.vt.edu), Nov 10 2004
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