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A000006 Integer part of square root of n-th prime.
(Formerly M0259 N0092)
15
1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Conjecture: No two successive terms in the sequence differ by more than 1. Proof of this would prove the converse of the theorem that every prime is surrounded by two consecutive squares, namely |sqrt(p)|^2 and (|sqrt(p)|+1)^2. - Cino Hilliard, Jan 22 2003

Equals the number of squares less than prime(n). Cf. A014689. - Zak Seidov Nov 04 2007

The above conjecture is Legendre's conjecture that for n > 0 there is always a prime between n^2 and (n+1)^2. See A014085, number of primes between two consecutive squares, which is also the number of times n is repeated in the present sequence. - Jean-Christophe Hervé, Oct 25 2013

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Matthew Parker, The first 100 million terms (7-Zip compressed file)

FORMULA

a(n) = A000196(A000040(n)). - Reinhard Zumkeller, Mar 24 2012

MATHEMATICA

a[n_] := IntegerPart[Sqrt[Prime[n]]]

IntegerPart[Sqrt[#]]&/@Prime[Range[80]] (* Harvey P. Dale, Mar 06 2012 *)

PROG

(PARI) a(n)=sqrtint(prime(n)); vector(100, n, a(n))

(PARI) apply(sqrtint, primes(100)) \\ Charles R Greathouse IV, Apr 26 2012

(Haskell)

a000006 = a000196 . a000040  -- Reinhard Zumkeller, Mar 24 2012

CROSSREFS

Cf. A014085.

See also A263846 (floor of cube root of prime(n)), A000196 (floor of sqrt(n)), A048766 (floor of cube root of n).

Sequence in context: A084557 A024417 A060021 * A061017 A248170 A225545

Adjacent sequences:  A000003 A000004 A000005 * A000007 A000008 A000009

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 28 17:57 EDT 2017. Contains 287241 sequences.