%I A008578
%S A008578 1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A008578 89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
%U A008578 181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271
%N A008578 Prime numbers at the beginning of the 20th century (today 1 is no longer 
               regarded as a prime).
%C A008578 The non-composite numbers.
%C A008578 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
%C A008578 Numbers n such that their largest divisor <= sqrt(n) equals 1. (See also 
               A161344, A161345, A161424). [From Omar E. Pol (info(AT)polprimos.com), 
               Jul 05 2009]
%C A008578 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 (2stepan(AT)rambler.ru), Sep 
               27 2009]
%C A008578 d(n)<3 [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 17 2009]
%D A008578 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.
%D A008578 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 11.
%D A008578 H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 
               17 (1995), no. 2, 64-68. Math. Rev. 96b:01035
%D A008578 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 A008578 D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie 
               Institute, Washington, D.C. 1909.
%D A008578 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
%D A008578 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
%H A008578 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica 
               de los numeros primos y perfectos</a> [From Omar E. Pol (info(AT)polprimos.com), 
               Jul 05 2009]
%H A008578 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polprdipi.jpg">
               Illustration: Divisors and pi(x)</a> [From Omar E. Pol (info(AT)polprimos.com), 
               Jul 05 2009]
%H A008578 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A008578 PrimeFan, <a href="http://primefan.tripod.com/Prime1ProCon.html">Arguments 
               for and against the primality of 1</a>.
%H A008578 G. P. Michon, <a href="http://home.att.net/~numericana/answer/numbers.htm#one">
               Is 1 a prime number ?</a>
%H A008578 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_convolution">
               Dirichlet convolution</a>
%H A008578 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv07.jpg">
               Illustration of initial terms</a> [From Omar E. Pol (info(AT)polprimos.com), 
               Oct 24 2009]
%H A008578 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv06.jpg">
               Illustration for A008578, A161344, A161345 and A161424</a> [From 
               Omar E. Pol (info(AT)polprimos.com), Oct 24 2009]
%F A008578 m is in the sequence iff sigma(m)+phi(m)=2m. - Farideh Firoozbakht (mymontain(AT)yahoo.com), 
               Jan 27 2005
%F A008578 a(n) = A158611(n+1) for n >= 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Jun 19 2009]
%F A008578 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.
%F A008578 This = A030014 * A008683. (Dirichlet convolution using offset 1 with 
               A030014)
%F A008578 This = A030013 * A000012. (Dirichlet convolution using offset 1 with 
               A030013)
%F A008578 This = A034773 * A007427. (Dirichlet convolution)
%F A008578 This = A034760 * A023900. (Dirichlet convolution)
%F A008578 This = A034762 * A046692. (Dirichlet convolution)
%F A008578 This * A000012 = A030014. (Dirichlet convolution using offset 1 with 
               A030014)
%F A008578 This * A008683 = A030013. (Dirichlet convolution using offset 1 with 
               A030013)
%F A008578 This * A000005 = A034773. (Dirichlet convolution)
%F A008578 This * A000010 = A034760. (Dirichlet convolution)
%F A008578 This * A000203 = A034762. (Dirichlet convolution)
%F A008578 A002033(a(n))=1. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), 
               Sep 27 2009]
%p A008578 A008578 := n->if n=1 then 1 else ithprime(i-1);
%t A008578 Join[ {1}, Table[ Prime[n], {n, 1, 60} ] ]
%Y A008578 See A000040, which is the main entry for this sequence. The complement 
               of A002808.
%Y A008578 Cf. A161344, A161345, A161424, A161835. [From Omar E. Pol (info(AT)polprimos.com), 
               Jul 05 2009]
%Y A008578 Cf. A002033. [From Juri-stepan Gerasimov (2stepan(AT)rambler.ru), Sep 
               27 2009]
%Y A008578 Cf. First column of array in A163280. Also, first row of array in A163990. 
               [From Omar E. Pol (info(AT)polprimos.com), Oct 24 2009]
%Y A008578 Sequence in context: A070159 A158611 A000040 this_sequence A100726 A015919 
               A064555
%Y A008578 Adjacent sequences: A008575 A008576 A008577 this_sequence A008579 A008580 
               A008581
%K A008578 nonn,easy,nice
%O A008578 1,2
%A A008578 N. J. A. Sloane (njas(AT)research.att.com).
%E A008578 Replaced a geocities.com URL - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 30 2009