cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A008578 Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime).

Original entry on oeis.org

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
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

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. - 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
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,...,n-1}. - 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.
  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 84 at pp. 214-217.
  • 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
  • H. C. Williams and J. O. Shallit, 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

Links

Crossrefs

The main entry for this sequence is A000040.
The complement of A002808.
Cf. A000732 (boustrophedon transform).
Cf. A000010, A000203.
Cf. A023626 (self-convolution).

Programs

  • GAP
    A008578:=Concatenation([1],Filtered([1..10^5],IsPrime)); # Muniru A Asiru, Sep 07 2017
  • Haskell
    a008578 n = a008578_list !! (n-1)
a008578_list = 1 : a000040_list
-- Reinhard Zumkeller, Nov 09 2011
  • Magma
    [1] cat [n: n in PrimesUpTo(271)];  // Bruno Berselli, Mar 05 2011
  • Maple
    A008578 := n->if n=1 then 1 else ithprime(n-1); 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[n-1], CoprimeQ[#, n]&];
Select[Range[271], oldPrimeQ] (* Jean-François Alcover, Jun 07 2017, after Peter Luschny *)
  • PARI
    is(n)=isprime(n)||n==1
  • Sage
    isA008578 = lambda n: all(gcd(k, n) == 1 for k in (1..n-1))
print([n for n in (1..271) if isA008578(n)]) # Peter Luschny, Jun 07 2017

Formula

a(n) = A000040(n-1).
m is in the sequence iff sigma(m) + phi(m) = A065387(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. - Juri-Stepan Gerasimov, Sep 27 2009
a(n) = A181363((2*n-1)*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))*(1-x)^2. - L. Edson Jeffery, Nov 25 2013

A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

Original entry on oeis.org

1, 1, 2, 3, 24, 5, 720, 105, 2240, 189, 3628800, 385, 479001600, 19305, 896896, 2027025, 20922789888000, 85085, 6402373705728000, 8729721, 47297536000, 1249937325, 1124000727777607680000, 37182145, 41363226782215962624, 608142583125, 1524503639859200000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

In other words, a(1) = 1 and for n >= 2, a(n) = product of the phi(n) numbers < n and relatively prime to n.
From Gauss's generalization of Wilson's theorem (see Weisstein link) it follows that, for n>1, a(n) == -1 (mod n) if and only if there exists a primitive root modulo n (cf. the Hardy and Wright reference, Theorem 129. p. 102). - Vladimir Shevelev, May 11 2012
Islam & Manzoor prove that a(n) is an injection for n > 1, see links. In other words, if a(m) = a(n), and min(m, n) > 1, then m = n. - Muhammed Hedayet, May 25 2016
Cosgrave & Dilcher propose the name Gauss factorial. Indeed the sequence is the special case N = n of the Gauss factorial N_n! = Product_{1<=j<=N, gcd(j, n)=1} j (see A216919). - Peter Luschny, Feb 07 2018

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A023896, A066570, A038566, A124441.
Main diagonal gives A216919.

Programs

  • Haskell
    a001783 = product . a038566_row
-- Reinhard Zumkeller, Mar 04 2012, Aug 26 2011
  • Maple
    A001783 := proc(n) local i,t1; t1 := 1; for i from 1 to n do if gcd(i,n)=1 then t1 := t1*i; fi; od; t1; end;
A001783 := proc(n) local i; mul(i,i=select(k->igcd(n,k)=1,[$1..n])) end; # Peter Luschny, Oct 30 2010
  • Mathematica
    A001783[n_]:=Times@@Select[Range[n],CoprimeQ[n,#]&];
Array[A001783,20] (* Enrique Pérez Herrero, Jul 23 2011 *)
  • PARI
    A001783(n)=prod(k=2,n-1,k^(gcd(k,n)==1))  \\ M. F. Hasler, Jul 23 2011
  • PARI
    a(n)=my(f=factor(n),t=n^eulerphi(f)); fordiv(f,d, t*=(d!/d^d)^moebius(n/d)); t \\ Charles R Greathouse IV, Nov 05 2015
  • Sage
    def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A001783(n): return Gauss_factorial(n, n)
[A001783(n) for n in (1..25)] # Peter Luschny, Oct 01 2012

Formula

a(n) = n^phi(n)*Product_{d|n} (d!/d^d)^mu(n/d); phi=A000010 is the Euler totient function and mu=A008683 the Moebius function (Tom M. Apostol, Introduction to Analytic Number Theory, New York 1984, p. 48). - Franz Vrabec, Jul 08 2005
a(n) = n!/A066570(n). - R. J. Mathar, Mar 10 2011
A001221(a(n)) = A000720(n) - A001221(n) = A048865(n).
A006530(a(n)) = A136548(n). - Enrique Pérez Herrero, Jul 23 2011
a(n) = A124441(n)*A124442(n). - M. F. Hasler, Jul 23 2011
a(n) == (-1)^A211487(n) (mod n). - Vladimir Shevelev, May 13 2012
a(n) = A250269(n) / A193679(n). - Daniel Suteu, Apr 05 2021

Extensions

More terms from James Sellers, Dec 23 1999

A305419 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over GF(2)[X], with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 13, 13, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 47, 47, 55, 55, 55, 55, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 67, 67, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 87
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2018

Keywords

Comments

For n >= 3, a(n) is the largest term of A014580 less than n.

Links

Crossrefs

Cf. A014580, A091225, A305420, A305422.
Cf. also A151799 (A136548), A305429.

Programs

  • PARI
    A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));

A305429 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over Q, with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 25, 25, 25, 25, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 55, 55, 55, 55, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 69, 69, 71, 71, 73, 73, 73, 73, 77, 77, 79, 79, 81, 81, 83, 83, 83, 83, 87, 87
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2018

Keywords

Comments

For n >= 3, a(n) is the largest term of A206074 less than n.

Links

Crossrefs

Cf. A206074, A257000, A305430.
Cf. also A151799 (A136548), A305419.

Programs

  • PARI
    A257000(n) = polisirreducible(Pol(binary(n)));
A305429(n) = if(n<3,1, my(k=n-1); while(k>1 && !A257000(k),k--); (k));

A096501 Difference between primes preceding n+1 and n.

Original entry on oeis.org

0, 4, 1, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 4, 0, 2, 0, 0
Offset: 1

Views

Author

Labos Elemer, Jul 09 2004

Keywords

Comments

Values a(1) = 0 and a(2) = 4 are based on convention in Mathematica-language that PreviousPrime(1) = PreviousPrime(2) = -2. - Antti Karttunen, Jan 03 2019

Links

Crossrefs

Cf. A000720, A001223, A010051, A096500, A136548, A151799, A175851, A007917.

Programs

Formula

For n > 2, a(n) = A010051(n) * A001223(A000720(n)-1) = A136548(1+n)-A136548(n). - Antti Karttunen, Jan 03 2019
a(n) = A007917(n) - A007917(n-1), for n > 2. - Ridouane Oudra, Oct 05 2024
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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