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A000267
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Integer part of square root of 4n+1.
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9
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1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 1^1, 2^1, 3^2, 4^2, 5^3, 6^3, 7^4, 8^4, 9^5, 10^5, ...
Start with n, repeatedly subtract the square root of the previous term; a(n) gives number of steps to reach 0. - Robert G. Wilson v, Jul 22, 2002.
a(n) = 1+a(n-[n^(1/2)]), n>0. - Michael Somos, Jul 22, 2002
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REFERENCES
| T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 20.
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 77, Entry 23.
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LINKS
| S. Ramanujan, Question 723, J. Ind. Math. Soc.
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FORMULA
| a(n) = int( 1 / ( sqrt(n + 1) - sqrt(n) ) ) - Robert A. Stump (bob_ess107(AT)yahoo.com), Apr 07 2003
a(n) = |{floor(n/k): k in Z+}| - David W. Wilson (davidwwilson(AT)comcast.net), May 26 2005
a(n) = ceil(2*sqrt(n+1) - 1). [Mircea Merca, Feb 03 2012]
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PROG
| (PARI) a(n)=if(n<0, 0, sqrtint(4*n+1))
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CROSSREFS
| [a(n)/2]=A000196(n). Cf. A080037.
Sequence in context: A086592 A132663 A023964 * A060020 A166127 A143502
Adjacent sequences: A000264 A000265 A000266 * A000268 A000269 A000270
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Michael Somos, Jun 13, 2000
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