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Search: id:A083515
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| A083515 |
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Smallest number k such that R(n) + k is a square, where R(n) = A002275. |
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+0
1
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| 0, 5, 10, 45, 125, 445, 1914, 4445, 1570, 44445, 156989, 444445, 941538, 4444445, 9826365, 44444445, 139362425, 444444445, 1287131805, 4444444445, 2222074045, 44444444445, 11388893810, 444444444445, 1138889380989, 4444444444445
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OFFSET
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1,2
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COMMENT
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For n > 1, a(n) is never zero. Proof: R(2) = 4*2+3, R(3) = 4*27+3. In general 111...1 = 4*2777...7+3, or if f(n) = 2777...7 = 7/9*(-1+10^n)+2^(n+1)*5^n, then R(n) = 4 * f(n-2) + 3. Hence repunits are of the form 4m+3 and cannot be square. --Paraphrased from Tattersall.
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REFERENCES
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J. Tattersall, "Elementary Number Theory in Nine Chapters". Cambridge University Press, 2001. pp. 57, 330.
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FORMULA
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a(n)=ceil(sqrt(R(n)))^2-R(n)
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EXAMPLE
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a(5) = 125 because 11111 + 125 = 11236 is a square.
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CROSSREFS
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Sequence in context: A038070 A136138 A122173 this_sequence A103971 A035406 A103932
Adjacent sequences: A083512 A083513 A083514 this_sequence A083516 A083517 A083518
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KEYWORD
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easy,nonn
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AUTHOR
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Jason Earls (zevi_35711(AT)yahoo.com), Jun 09 2003
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