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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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182
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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 (info(AT)polprimos.com), 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 (2stepan(AT)rambler.ru), Sep 27 2009]
d(n)<3 [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 17 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 (jaroslav.krizek(AT)atlas.cz), Mar 01 2010]
Where record values of A022404 occur: A086332(n)=A022404(a(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 21 2010]
a(n) = A181363((2*n-1)*2^k), k >= 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 16 2010]
1 together with the prime numbers. - Omar E. Pol, Mar 04 2011
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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. 870.
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
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
| 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, Determinacion geometrica de los numeros primos y perfectos [From Omar E. Pol (info(AT)polprimos.com), Jul 05 2009]
O. E. Pol, Illustration: Divisors and pi(x) [From Omar E. Pol (info(AT)polprimos.com), Jul 05 2009]
O. E. Pol, Illustration of initial terms [From Omar E. Pol (info(AT)polprimos.com), Oct 24 2009]
O. E. Pol, Illustration for A008578, A161344, A161345 and A161424 [From Omar E. Pol (info(AT)polprimos.com), Oct 24 2009]
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
| m is in the sequence iff sigma(m)+phi(m)=2m. - Farideh Firoozbakht (mymontain(AT)yahoo.com), Jan 27 2005
a(n) = A158611(n+1) for n >= 1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 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 (2stepan(AT)rambler.ru), 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
| A008578 := n->if n=1 then 1 else ithprime(n-1); fi :
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MATHEMATICA
| Join[ {1}, Table[ Prime[n], {n, 1, 60} ] ]
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PROG
| (PARI) a(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
| See A000040, which is the main entry for this sequence. The complement of A002808.
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]
Sequence in context: A158611 A182986 A000040 * A100726 A015919 A064555
Adjacent sequences: A008575 A008576 A008577 * A008579 A008580 A008581
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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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