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A008578
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Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime).
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202
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1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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The noncomposite numbers.
Also smallest sequence with the property that the product of 2 or more elements with different indices is never a square. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
Numbers n such that their largest divisor <= sqrt(n) equals 1. (See also A161344, A161345, A161424). [From Omar E. Pol, Jul 05 2009]
Or numbers n with only perfect partition; also numbers such that 1=number of perfect partitions of n; or, unit together with the prime numbers A000040. [From Juri-Stepan Gerasimov, Sep 27 2009]
Numbers n such that d(n) < 3. [From Juri-Stepan Gerasimov, Oct 17 2009]
Also first column of array in A163280. Also first row of array in A163990. - Omar E. Pol, Oct 24 2009
a(n) = possible values of A136548(m) in increasing order, where A136548(m) = the largest numbers h such that A000203(h) <= k (k = 1,2,3,..), where A000203(h) = sum of divisors of h. [From Jaroslav Krizek, Mar 01 2010]
Where record values of A022404 occur: A086332(n)=A022404(a(n)). [From Reinhard Zumkeller, Jun 21 2010]
a(n) = A181363((2*n-1)*2^k), k >= 0. [From Reinhard Zumkeller, Oct 16 2010]
1 together with the prime numbers. - Omar E. Pol, Mar 04 2011
A060448(a(n)) = 1. [Reinhard Zumkeller, Apr 05 2012]
Positive integers that have no divisors other than 1 and itself (the old definition of prime numbers). - Omar E. Pol, Aug 10 2012
A086971(a(n)) = 0. - Reinhard Zumkeller, Dec 14 2012
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
C. K. Caldwell and Y. Xiong, What is the smallest prime?, arXiv preprint arXiv:1209.2007, 2012. - From N. J. A. Sloane, Dec 26 2012
G. Chrystal, Algebra: An Elementary Textbook. Chelsea Publishing Company, 7th edition, (1964), chap. III.7, p. 38.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 11.
H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035
D. H. Lehmer, The sieve problem for all-purpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 6-14. Math. Rev. 14:691e
D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie Institute, Washington, D.C. 1909.
R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082
A. Reddick and Y. Xiong, The search for one as a prime number: from ancient Greece to modern times, Electronic Journal of Undergraduate Mathematics, Volume 16, 1 { 13, 2012. - From N. J. A. Sloane, Feb 03 2013
Williams, H. C.; Shallit, J. O. Factoring integers before computers. Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), 481-531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143
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LINKS
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Charles R Greathouse IV, 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]
G. P. Michon, Is 1 a prime number?
O. E. Pol, Illustration of initial terms
O. E. Pol, Illustration of initial terms of A008578, A161344, A161345, A161424
PrimeFan, Arguments for and against the primality of 1
J. Todd, Review of Lehmer's tables, Mathematical Tables and Other Aids.., Vol. 11, No. 60, (1957) (on JSTOR.org).
Wikipedia, Dirichlet convolution
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FORMULA
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m is in the sequence iff sigma(m)+phi(m)=2m. - Farideh Firoozbakht, Jan 27 2005
a(n) = A158611(n+1) for n >= 1. [From Jaroslav Krizek, Jun 19 2009]
In the following formulas (based on emails from Jaroslav Krizek and R. J. Mathar), the star denotes a Dirichlet convolution between two sequences, and "This" is A008578.
This = A030014 * A008683. (Dirichlet convolution using offset 1 with A030014)
This = A030013 * A000012. (Dirichlet convolution using offset 1 with A030013)
This = A034773 * A007427. (Dirichlet convolution)
This = A034760 * A023900. (Dirichlet convolution)
This = A034762 * A046692. (Dirichlet convolution)
This * A000012 = A030014. (Dirichlet convolution using offset 1 with A030014)
This * A008683 = A030013. (Dirichlet convolution using offset 1 with A030013)
This * A000005 = A034773. (Dirichlet convolution)
This * A000010 = A034760. (Dirichlet convolution)
This * A000203 = A034762. (Dirichlet convolution)
A002033(a(n))=1. [From Juri-Stepan Gerasimov, Sep 27 2009]
A033273(a(n))=1 [From Juri-Stepan(AT)rambler.ru (2stepan(AT)rambler.ru), Dec 07 2009]
a(n) = A001747(n)/2. - Omar E. Pol, Jan 30 2012
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MAPLE
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A008578 := n->if n=1 then 1 else ithprime(n-1); fi :
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MATHEMATICA
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Join[ {1}, Table[ Prime[n], {n, 1, 60} ] ]
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PROG
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(PARI) is(n)=isprime(n)||n==1
(MAGMA) [1] cat [n: n in PrimesUpTo(271)]; // Bruno Berselli, Mar 05 2011
(Haskell)
a008578 n = a008578_list !! (n-1)
a008578_list = 1 : a000040_list
-- Reinhard Zumkeller, Nov 09 2011
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CROSSREFS
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Main entry for this sequence is A000040. The complement of A002808.
Sequence in context: A216883 A216884 A216885 A216886 A100726 A015919 A064555
Adjacent sequences: A008575 A008576 A008577 * A008579 A008580 A008581
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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