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 1-6 of 6 results.

A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090
Offset: 0

Views

Author

N. J. A. Sloane and J. H. Conway

Keywords

Comments

See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.
a(n) is the least number N with n distinct prime factors (i.e., omega(N) = n, cf. A001221). - Lekraj Beedassy, Feb 15 2002
Phi(n)/n is a new minimum for each primorial. - Robert G. Wilson v, Jan 10 2004
Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - Lekraj Beedassy, Mar 31 2005
Apparently each term is a new minimum for phi(x)*sigma(x)/x^2. 6/Pi^2 < sigma(x)*phi(x)/x^2 < 1 for n > 1. - Jud McCranie, Jun 11 2005
Let f be a multiplicative function with f(p) > f(p^k) > 1 (p prime, k > 1), f(p) > f(q) > 1 (p, q prime, p < q). Then the record maxima of f occur at n# for n >= 1. Similarly, if 0 < f(p) < f(p^k) < 1 (p prime, k > 1), 0 < f(p) < f(q) < 1 (p, q prime, p < q), then the record minima of f occur at n# for n >= 1. - David W. Wilson, Oct 23 2006
Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
Records in number of distinct prime divisors. - Artur Jasinski, Apr 06 2008
For n >= 2, the digital roots of a(n) are multiples of 3. - Parthasarathy Nambi, Aug 19 2009 [with corrections by Zak Seidov, Aug 30 2015]
Denominators of the sum of the ratios of consecutive primes (see A094661). - Vladimir Joseph Stephan Orlovsky, Oct 24 2009
Where record values occur in A001221. - Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010
It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i = 0..T(sqrt(N))} A005867(i)/A002110(i). This can show for example that at least 0.16*N numbers are primes less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010
The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. prime(n)# is divisible by 3 for n >= 2. - Christian Schulz, Oct 30 2013
The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "near-primorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3. - Richard R. Forberg, Jul 01 2015
If a number k and a(n) are coprime and k < (prime(n+1))^b < a(n), where b is an integer, then k has fewer than b prime factors, counting multiplicity (i.e., bigomega(k) < b, cf. A001222). - Isaac Saffold, Dec 03 2017
If n > 0, then a(n) has 2^n unitary divisors (A034444), and a(n) is a record; i.e., if k < a(n) then k has fewer unitary divisors than a(n) has. - Clark Kimberling, Jun 26 2018
Unitary superabundant numbers: numbers k with a record value of the unitary abundancy index, A034448(k)/k > A034448(m)/m for all m < k. - Amiram Eldar, Apr 20 2019
Psi(n)/n is a new maximum for each primorial (psi = A001615) [proof in link: Patrick Sole and Michel Planat, proposition 1 page 2]; compare with comment 2004: Phi(n)/n is a new minimum for each primorial. - Bernard Schott, May 21 2020
The term "primorial" was coined by Harvey Dubner (1987). - Amiram Eldar, Apr 16 2021
a(n)^(1/n) is approximately (n log n)/e. - Charles R Greathouse IV, Jan 03 2023
Subsequence of A267124. - Frank M Jackson, Apr 14 2023

Examples

			a(9) = 23# = 2*3*5*7*11*13*17*19*23 = 223092870 divides the difference 5283234035979900 in the arithmetic progression of 26 primes A204189. - _Jonathan Sondow_, Jan 15 2012

References

  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 49.
  • P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
  • 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).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.
  • D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.

Links

Crossrefs

A034386 gives the second version of the primorial numbers.
Subsequence of A005117 and of A064807. Apart from the first term, a subsequence of A083207.
Cf. A001615, A002182, A002201, A003418, A005235, A006862, A034444 (unitary divisors), A034448, A034387, A033188, A035345, A035346, A036691 (compositorial numbers), A049345 (primorial base representation), A057588, A060735 (and integer multiples), A061742 (squares), A072938, A079266, A087315, A094348, A106037, A121572, A053589, A064648, A132120, A260188.
Cf. A061720 (first differences), A143293 (partial sums).
Cf. also A276085, A276086.
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

Programs

  • Haskell
    a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
-- Reinhard Zumkeller, Feb 19 2012, May 03 2011
  • Magma
    [1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012
  • Magma
    [1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015
  • Maple
    A002110 := n -> mul(ithprime(i),i=1..n);
  • Mathematica
    FoldList[Times, 1, Prime[Range[20]]]
primorial[n_] := Product[Prime[i], {i, n}]; Array[primorial,20] (* José María Grau Ribas, Feb 15 2010 *)
Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* Harvey P. Dale, Apr 11 2012 *)
  • PARI
    a(n)=prod(i=1,n, prime(i)) \\ Washington Bomfim, Sep 23 2008
  • PARI
    p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) )  \\ Harry J. Smith, Nov 13 2009
  • PARI
    a(n) = factorback(primes(n)) \\ David A. Corneth, May 06 2018
  • Python
    from sympy import primorial
def a(n): return 1 if n < 1 else primorial(n)
[a(n) for n in range(51)]  # Indranil Ghosh, Mar 29 2017
  • Sage
    [sloane.A002110(n) for n in (1..20)] # Giuseppe Coppoletta, Dec 05 2014
  • Scheme
    ; with memoization-macro definec
(definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 (- n 1))))) ;; Antti Karttunen, Aug 30 2016

Formula

Asymptotic expression for a(n): exp((1 + o(1)) * n * log(n)) where o(1) is the "little o" notation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
a(n) = A054842(A002275(n)).
Binomial transform = A136104: (1, 3, 11, 55, 375, 3731, ...). Equals binomial transform of A121572: (1, 1, 3, 17, 119, 1509, ...). - Gary W. Adamson, Dec 14 2007
a(0) = 1, a(n+1) = prime(n)*a(n). - Juri-Stepan Gerasimov, Oct 15 2010
a(n) = Product_{i=1..n} A000040(i). - Jonathan Vos Post, Jul 17 2008
a(A051838(n)) = A116536(n) * A007504(A051838(n)). - Reinhard Zumkeller, Oct 03 2011
A000005(a(n)) = 2^n. - Carlos Eduardo Olivieri, Jun 16 2015
a(n) = A035345(n) - A005235(n) for n > 0. - Jonathan Sondow, Dec 02 2015
For all n >= 0, a(n) = A276085(A000040(n+1)), a(n+1) = A276086(A143293(n)). - Antti Karttunen, Aug 30 2016
A054841(a(n)) = A002275(n). - Michael De Vlieger, Aug 31 2016
a(n) = A270592(2*n+2) - A270592(2*n+1) if 0 <= n <= 4 (conjectured for all n by Alon Kellner). - Jonathan Sondow, Mar 25 2018
Sum_{n>=1} 1/a(n) = A064648. - Amiram Eldar, Oct 16 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = A132120. - Amiram Eldar, Apr 12 2021
Theta being Chebyshev's theta function, a(0) = exp(theta(1)), and for n > 0, a(n) = exp(theta(m)) for A000040(n) <= m < A000040(n+1) where m is an integer. - Miles Englezou, Nov 26 2024

A005235 Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.

Original entry on oeis.org

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Reo F. Fortune conjectured that a(n) is always prime.
You might be searching for Fortunate Primes, which is an alternative name for this sequence. It is not the official name yet, because it is possible, although unlikely, that not all the terms are primes. - N. J. A. Sloane, Sep 30 2020
The first 500 terms are primes. - Robert G. Wilson v. The first 2000 terms are primes. - Joerg Arndt, Apr 15 2013
The strong form of Cramér's conjecture implies that a(n) is a prime for n > 1618, as previously noted by Golomb. - Charles R Greathouse IV, Jul 05 2011
a(n) is the smallest m such that m > 1 and A002110(n) + m is prime. For every n, a(n) must be greater than prime(n+1) - 1. - Farideh Firoozbakht, Aug 20 2003
If a(n) < prime(n+1)^2 then a(n) is prime. According to Cramér's conjecture a(n) = O(prime(n)^2). - Thomas Ordowski, Apr 09 2013
Conjectures from Pierre CAMI, Sep 08 2017: (Start)
If all terms are prime, then lim_{N->oo} (Sum_{n=1..N} primepi(a(n))) / (Sum_{n=1..N} n) = 3/2, and primepi(a(n))/n < 6 for all n.
Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = Pi/2.
a(n)/prime(n) < 8 for all n. (End)
Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 3/2. - Alain Rocchelli, Dec 24 2022
The name "Fortunate numbers" was coined by Golomb (1981) after the New Zealand social anthropologist Reo Franklin Fortune (1903 - 1979). According to Golomb, Fortune's conjecture first appeared in print in Martin Gardner's Mathematical Games column in 1980. - Amiram Eldar, Aug 25 2020

Examples

			a(4) = 13 because P_4# = 2*3*5*7 = 210, plus one is 211, the next prime is 223 and the difference between 210 and 223 is 13.

References

  • Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially pp. 194-195.
  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 1994, Section A2, p. 11.
  • Stephen P. Richards, A Number For Your Thoughts, 1982, p. 200.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 114-115.
  • David Wells, Prime Numbers: The Most Mysterious Figures In Math, Hoboken, New Jersey: John Wiley & Sons (2005), pp. 108-109.

Links

Crossrefs

Cf. A046066, A002110, A006862, A035345, A035346, A055211, A129912, A010051, A005408, A038771, A038711.

Programs

  • Haskell
    a005235 n = head [m | m <- [3, 5 ..], a010051'' (a002110 n + m) == 1]
-- Reinhard Zumkeller, Apr 02 2014
  • Maple
    Primorial:= 2:
p:= 2:
A[1]:= 3:
for n from 2 to 100 do
  p:= nextprime(p);
  Primorial:= Primorial * p;
  A[n]:= nextprime(Primorial+p+1)-Primorial;
od:
seq(A[n],n=1..100); # Robert Israel, Dec 02 2015
  • Mathematica
    NPrime[n_Integer] := Module[{k}, k = n + 1; While[! PrimeQ[k], k++]; k]; Fortunate[n_Integer] := Module[{p, q}, p = Product[Prime[i], {i, 1, n}] + 1; q = NPrime[p]; q - p + 1]; Table[Fortunate[n], {n, 60}]
r[n_] := (For[m = (Prime[n + 1] + 1)/2, ! PrimeQ[Product[Prime[k], {k, n}] + 2 m - 1], m++]; 2 m - 1); Table[r[n], {n, 60}]
FN[n_] := Times @@ Prime[Range[n]]; Table[NextPrime[FN[k] + 1] - FN[k], {k, 60}] (* Jayanta Basu, Apr 24 2013 *)
NextPrime[#]-#+1&/@(Rest[FoldList[Times,1,Prime[Range[60]]]]+1) (* Harvey P. Dale, Dec 15 2013 *)
  • PARI
    a(n)=my(P=prod(k=1,n,prime(k)));nextprime(P+2)-P \\ Charles R Greathouse IV, Jul 15 2011; corrected by Jean-Marc Rebert, Jul 28 2015
  • Python
    from sympy import nextprime, primorial
def a(n): psharp = primorial(n); return nextprime(psharp+1) - psharp
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Jan 15 2022
  • Sage
    def P(n): return prod(nth_prime(k) for k in range(1, n + 1))
it = (P(n) for n in range(1, 31))
print([next_prime(Pn + 2) - Pn for Pn in it]) # F. Chapoton, Apr 28 2020

Formula

If x(n) = 1 + Product_{i=1..n} prime(i), q(n) = least prime > x(n), then a(n) = q(n) - x(n) + 1.
a(n) = 1 + the difference between the n-th primorial plus one and the next prime.
a(n) = A035345(n) - A002110(n). - Jonathan Sondow, Dec 02 2015

A265109 a(n) = largest k such that prime(n) + A002110(k) is prime.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 6, 7, 8, 6, 5, 9, 8, 8, 14, 9, 16, 17, 14, 14, 16, 21, 11, 19, 14, 24, 25, 15, 18, 11, 28, 12, 8, 19, 16, 22, 35, 31, 36, 25, 31, 16, 40, 30, 23, 41, 39, 35, 10, 32, 43, 38, 24, 41, 19, 35, 23, 55, 54, 24, 53, 50, 57, 62, 48, 36, 64, 21, 45, 54
Offset: 1

Views

Author

Altug Alkan, Dec 01 2015

Keywords

Comments

a(n) = n-1 iff a(n) is in A035346 for n > 1.

Examples

			a(4) = 3 because A002110(3) + prime(4) = A002110(3) + 7 = 37 is prime.

Links

Crossrefs

Cf. A000040, A002110, A035346, A100380.

Programs

  • PARI
    a(n) = {my(k=1); while(k, if(ispseudoprime(prod(i=1, n-k, prime(i)) + prime(n)), return(n-k)); k++)}
  • Python
    from itertools import count
from sympy import isprime, prime, primorial
def A002110(n): return primorial(n) if n > 0 else 1
def A265109(n):
    pn = prime(n)
    return next(k for k in range(n-1, -1, -1) if isprime(pn+A002110(k)))
print([A265109(n) for n in range(1, 31)]) # Michael S. Branicky, Jan 10 2025

Formula

a(n) < n.

A038708 Primes of the form (k-th primorial) + (k+1)-st prime.

Original entry on oeis.org

5, 11, 37, 30047, 510529, 9699713, 13082761331670077, 32589158477190044789, 1922760350154212639131, 40729680599249024150621323549
Offset: 1

Views

Author

Labos Elemer, May 02 2000

Keywords

Examples

			k=6: 6th primorial 2*3*5*7*11*13 + prime(7) = 300030 + 17 = 30047 is a prime.

Links

Crossrefs

Cf. A002110, A035346, A005235.

Programs

  • Mathematica
    With[{nn=30},Select[Total/@Thread[{FoldList[Times,Prime[ Range[ nn]]],Prime[ Range[ 2,nn+1]]}],PrimeQ]] (* Harvey P. Dale, Oct 12 2018 *)

Extensions

Next term has 149 decimal digits.

A093077 Primes p = prime(i) such that p(i)# + p(i+1) is prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 19, 43, 53, 59, 73, 367, 6143
Offset: 1

Views

Author

Robert G. Wilson v, Oct 25 2003

Keywords

Comments

Prime(A035346(n)) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 19 2004

Examples

			3 = p(2) is in the sequence because p(2)# + p(3) = 11 is prime.

Links

Crossrefs

Cf. A087714, A088415, A093078.

Programs

  • Mathematica
    Do[p = Product[ Prime[i], {i, 1, n}]; q = Prime[n + 1]; If[ PrimeQ[p + q], Print[ Prime[n]]], {n, 1, 1435}]
With[{nn=1000},NextPrime[#,-1]&/@(Select[Thread[{FoldList[ Times,Prime[ Range[nn]]],Prime[Range[ 2,nn+1]]}], PrimeQ[ Total[#]]&][[All,2]])] (* Harvey P. Dale, Oct 07 2018 *)

A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 11, 13, 31, 23, 19, 37, 41, 29, 31, 43, 53, 41, 53, 79, 59, 97, 59, 61, 113, 97, 179, 73, 73, 97, 103, 101, 109, 101, 229, 109, 139, 113, 227, 131, 191, 163, 139, 199, 151, 139, 181, 223, 229, 367, 239, 499, 251, 509, 251, 227, 373, 281, 233
Offset: 1

Views

Author

Labos Elemer, Nov 14 2000

Keywords

Comments

Analogous to Fortunate numbers and like them so far proved to be primes. This holds for x<=421: if Q is the first follower prime, then Q(421)-lcm(1,...421) = 557. For first some cases when 1+LCM is also a prime, the 2nd primes give 3,5,5,7,11,11,.. deviations, i.e. give primes.

Crossrefs

Cf. A000961, A003418, A051451, A057019, A037153, A035346, A005235, A054272, A055211, A037155, A045493, A038710 etc

Programs

  • PARI
    N=1; for(n=2,1e3, if(isprimepower(n,&p), N*=p; print1(nextprime(N+2)-N", "))) \\ Charles R Greathouse IV, Nov 18 2015

Extensions

Name corrected by Charles R Greathouse IV, Nov 18 2015
Showing 1-6 of 6 results.

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