

A008578


Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime).


505



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
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OFFSET

1,2


COMMENTS

1 together with the primes; also called the noncomposite numbers.
Also largest sequence of nonnegative integers 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 [Comment corrected by Farideh Firoozbakht, Aug 03 2014]
Numbers k whose largest divisor <= sqrt(k) equals 1. (See also A161344, A161345, A161424.)  Omar E. Pol, Jul 05 2009
Numbers k such that d(k) <= 2.  JuriStepan Gerasimov, Oct 17 2009
Also first column of array in A163280. Also first row of array in A163990.  Omar E. Pol, Oct 24 2009
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.  Jaroslav Krizek, Mar 01 2010
Where record values of A022404 occur: A086332(n)=A022404(a(n)).  Reinhard Zumkeller, Jun 21 2010
Positive integers that have no divisors other than 1 and itself (the old definition of prime numbers).  Omar E. Pol, Aug 10 2012
Conjecture: the sequence contains exactly those k such that sigma(k) > k*BigOmega(k).  Irina Gerasimova, Jun 08 2013
Note on the Gerasimova conjecture: all terms in the sequence obviously satisfy the inequality, because sigma(p) = 1+p and BigOmega(p) = 1 for primes p, so 1+p > p*1. For composites, the (opposite) inequality is heuristically correct at least up to k <= 4400000. The general proof requires to show that BigOmega(k) is an upper limit of the abundancy sigma(k)/k for composite k. This proof is easy for semiprimes k=p1*p2 in general, where sigma(k)=1+p1+p2+p1*p2 and BigOmega(k)=2 and p1, p2 <= 2.  R. J. Mathar, Jun 12 2013
Numbers k such that phi(k) + sigma(k) = 2k.  Farideh Firoozbakht, Aug 03 2014
isA008578(n) <=> k is prime to n for all k in {1,2,...,n1}.  Peter Luschny, Jun 05 2017
In 1751 Leonhard Euler wrote: "Having so established this sign S to indicate the sum of the divisors of the number in front of which it is placed, it is clear that, if p indicates a prime number, the value of Sp will be 1 + p, except for the case where p = 1, because then we have S1 = 1, and not S1 = 1 + 1. From this we see that we must exclude unity from the sequence of prime numbers, so that unity, being the start of whole numbers, it is neither prime nor composite."  Omar E. Pol, Oct 12 2021
a(1) = 1; for n >= 2, a(n) is the least unused number that is coprime to all previous terms.  Jianing Song, May 28 2022
A number p is preprime if p = a*b ==> a = 1 or b = 1. This sequence lists the preprimes in the commutative monoid IN \ {0}.  Peter Luschny, Aug 26 2022


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 [19051991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 6468. Math. Rev. 96b:01035
D. H. Lehmer, The sieve problem for allpurpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 614. 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, 113139. Math. Rev. 96m:11082
Williams, H. C.; Shallit, J. O. Factoring integers before computers. Mathematics of Computation 19431993: a halfcentury of computational mathematics (Vancouver, BC, 1993), 481531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143


LINKS

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]
Chris K. Caldwell, Angela Reddick, Yeng Xiong and Wilfrid Keller, The History of the Primality of One: A Selection of Sources, (a dynamic survey), Journal of Integer Sequences, Vol. 15 (2012), #12.9.8.
C. K. Caldwell and Y. Xiong, What is the smallest prime?, arXiv preprint arXiv:1209.2007 [math.HO], 2012, and J. Int. Seq. 15 (2012) #12.9.6
Leonhard Euler, Découverte d’une loi tout extraordinaire des nombres, par rapport à la somme de leurs diviseurs, in Bibliothèque impartiale, 3, 1751, pp. 1031. Reprinted in Opera Postuma, 1, 1862, p.7684. Number 175 in the Eneström index.
G. P. Michon, Is 1 a prime number?
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of initial terms of A008578, A161344, A161345, A161424
PrimeFan, Arguments for and against the primality of 1
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
J. Todd, Review of Lehmer's tables, Mathematical Tables and Other Aids to Computation, Vol. 11, No. 60, (1957) (on JSTOR.org).
Wikipedia, Complete sequence.
Wikipedia, Dirichlet convolution


FORMULA

m is in the sequence iff sigma(m) + phi(m) = 2m.  Farideh Firoozbakht, Jan 27 2005
a(n) = A158611(n+1) for n >= 1.  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.  JuriStepan Gerasimov, Sep 27 2009
a(n) = A181363((2*n1)*2^k), k >= 0.  Reinhard Zumkeller, Oct 16 2010
a(n) = A001747(n)/2.  Omar E. Pol, Jan 30 2012
A060448(a(n)) = 1.  Reinhard Zumkeller, Apr 05 2012
A086971(a(n)) = 0.  Reinhard Zumkeller, Dec 14 2012
Sum_{n>=1} x^a(n) = (Sum_{n>=1} (A002815(n)*x^n))*(1x)^2.  L. Edson Jeffery, Nov 25 2013


MAPLE

A008578 := n>if n=1 then 1 else ithprime(n1); fi :


MATHEMATICA

Join[ {1}, Table[ Prime[n], {n, 1, 60} ] ]
NestList[ NextPrime, 1, 57] (* Robert G. Wilson v, Jul 21 2015 *)
oldPrimeQ[n_] := AllTrue[Range[n1], CoprimeQ[#, n]&];
Select[Range[271], oldPrimeQ] (* JeanFrançois Alcover, Jun 07 2017, after Peter Luschny *)


PROG

(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 !! (n1)
a008578_list = 1 : a000040_list
 Reinhard Zumkeller, Nov 09 2011
(Sage)
isA008578 = lambda n: all(gcd(k, n) == 1 for k in (1..n1))
print([n for n in (1..271) if isA008578(n)]) # Peter Luschny, Jun 07 2017
(GAP)
A008578:=Concatenation([1], Filtered([1..10^5], IsPrime)); # Muniru A Asiru, Sep 07 2017


CROSSREFS

The main entry for this sequence is A000040.
The complement of A002808.
Cf. A000732 (boustrophedon transform).
Cf. A000010, A000203.
Cf. A023626 (selfconvolution).
Sequence in context: A226159 A182986 A000040 * A216883 A216884 A216885
Adjacent sequences: A008575 A008576 A008577 * A008579 A008580 A008581


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



