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Search: id:A104272
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| A104272 |
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Ramanujan primes: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes < = x. |
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14
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| 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 569, 571, 587, 593, 599, 601, 607, 641, 643, 647, 653, 659
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Referring to his proof of Bertrand's postulate, Ramanujan states a generalization: "From this we easily deduce that pi(x) - pi(x/2) >= 1, 2, 3, 4, 5, ..., if x >= 2, 11, 17, 29, 41, ..., respectively." Since the a(n) are prime (by their minimality), I call them "Ramanujan primes."
See the additional references and links mentioned in A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
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REFERENCES
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S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc. 11 (1919), 181-182.
S. Ramanujan, Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara, and B. M. Wilson), Amer. Math. Soc., Providence, 2000, pp. 208-209.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
S. Ramanujan, A Proof Of Bertrand's Postulate
Eric Weisstein's World of Mathematics, Bertrand's Postulate
Eric Weisstein's World of Mathematics, Ramanujan Prime
Wikipedia, Ramanujan prime
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FORMULA
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a(n) = 1 + max{k: pi(k) - pi(k/2) = n - 1}.
a(n) = A080360(n-1) + 1 for n > 1 [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 11 2008]
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EXAMPLE
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a(1) = 2 is Bertrand's postulate: pi(x) - pi(x/2) >= 1 for all x >= 2.
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CROSSREFS
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Cf. A006992 Bertrand primes, A056171 pi(n) - pi(n/2).
Sequence in context: A066794 A087379 A019364 this_sequence A117155 A141176 A118839
Adjacent sequences: A104269 A104270 A104271 this_sequence A104273 A104274 A104275
Cf. A000720, A014085, A060715, A143223, A143224, A143225, A143226, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
Cf. A080360. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 11 2008]
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KEYWORD
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nonn,nice,new
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AUTHOR
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Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Feb 27 2005
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