A007014 -id:A007014 - cp's OEIS frontend

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

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

Offset: 0

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

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

			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

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.

Alex Ermolaev, Table of n, a(n) for n = 0..350 (terms up to a(100) from T. D. Noe)
Iskander Aliev, Jesús De Loera, Fritz Eisenbrand, Timm Oertel, and Robert Weismantel, The Support of Integer Optimal Solutions, arXiv:1712.08923 [math.OC], 2017.
C. K. Caldwell, The Prime Glossary, Primorial.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
Harvey Dubner, Factorial and primorial primes, J. Rec. Math., Vol. 19, No. 3 (1987), pp. 197-203. (Annotated scanned copy)
F. Ellermann, Illustration for A002110, A005867, A038110, A060753.
S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
D. J. Greenhoe, MRA-Wavelet subspace architecture for logic, probability, and symbolic sequence processing, 2014.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, pp. 7, 81.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 7.
A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, preprint, CONCUR 2014 - Concurrency Theory, Volume 8704 of the series Lecture Notes in Computer Science pp. 327-341.
A. W. Lin and S. Zhou, A linear-time algorithm for the orbit problem over cyclic groups, CONCUR 2014 - Concurrency Theory, Lecture Notes in Computer Science, Volume 8704, 2014, pp. 327-341.
F. E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012.
Thomas Morrill, Further Development of "Non-Pythagorean" Musical Scales Based on Logarithms, arXiv:1804.08067 [math.HO], 2018.
J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.
Patrick Sole and Michel Planat, The Robin inequality for 7-free integers, INTEGERS, 2011, #A65.
Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.
G. Villemin's Almanach of Numbers, Primorielle.
Eric Weisstein's World of Mathematics, Primorial.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

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.

a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
-- Reinhard Zumkeller, Feb 19 2012, May 03 2011

[1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012

[1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015

A002110 := n -> mul(ithprime(i),i=1..n);

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 *)

a(n)=prod(i=1,n, prime(i)) \\ Washington Bomfim, Sep 23 2008

p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) )  \\ Harry J. Smith, Nov 13 2009

a(n) = factorback(primes(n)) \\ David A. Corneth, May 06 2018

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

[sloane.A002110(n) for n in (1..20)] # Giuseppe Coppoletta, Dec 05 2014

; with memoization-macro definec
(definec (A002110 n) (if (zero? n) 1 (* (A000040 n) (A002110 (- n 1))))) ;; Antti Karttunen, Aug 30 2016

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

A003604 Number of primes <= n!.

Original entry on oeis.org

0, 0, 1, 3, 9, 30, 128, 675, 4231, 30969, 258689, 2428956, 25306287, 289620751, 3610490805, 48686912930, 706003798139, 10953617995740, 181035032207760, 3175094503778521, 58893601709552538, 1151825702178908788, 23688535118132456668, 511050155316058710033

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Number of distinct prime divisors of (n!)!, (A000197). - Jason Earls, Jul 04 2001

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Baugh, Table of n, a(n) for n = 0..25 (a(24)-a(25) found using Kim Walisch's primecount program).
Andrew R. Booker, The Nth Prime Page
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993
R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993
R. G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994
Index entries for sequences related to numbers of primes in various ranges

Cf. A000040, A000197.

Mathematica
```
Table[PrimePi[n!], {n, 0, 16}]
```
PARI
```
for(n=0,10,print(omega(n!!)))
```

a(n)=primepi(n!) \\ Charles R Greathouse IV, Jan 21 2016

[prime_pi(factorial(n)) for n in range(0, 14)] # Zerinvary Lajos, Jun 06 2009

a(15) from Jud McCranie
a(16)-a(17) from Paul Zimmermann
a(18) from Donovan Johnson, Dec 18 2009
a(19) from Donovan Johnson, Feb 18 2010
a(20) from Henri Lifchitz, Nov 11 2012
a(21)-a(23) from Henri Lifchitz, Aug 26 2014

A038710 a(n) is the smallest prime > product of the first n primes (A002110(n)).

Original entry on oeis.org

2, 3, 7, 31, 211, 2311, 30047, 510529, 9699713, 223092907, 6469693291, 200560490131, 7420738134871, 304250263527281, 13082761331670077, 614889782588491517, 32589158477190044789, 1922760350154212639131, 117288381359406970983379, 7858321551080267055879179

Offset: 0

Labos Elemer, May 02 2000

nonn

			for n=1,2,3,4,5,11,75, A002110(n)+1 gives smaller primes than A002110(n)+p, where p is a fortunate number (prime). At n=5, both 2311 and 2333 are primes but the first is smaller.

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A002110, A005235, A007014, A018239, A035345, A038711.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= n-> nextprime(p(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 16 2020

nmax = 2^16384; npd = 1; n = 1; npd = npd*Prime[n]; While[npd < nmax, cp = npd + 1; While[ ! (PrimeQ[cp]), cp = cp + 2]; Print[cp]; n = n + 1; npd = npd*Prime[n]] (* Lei Zhou, Feb 15 2005 *)
NextPrime/@FoldList[Times,1,Prime[Range[25]]] (* Harvey P. Dale, Dec 17 2010 *)

a(n) = nextprime(1+factorback(primes(n))); \\ Michel Marcus, Sep 25 2016; Dec 24 2022

from sympy import nextprime, primorial
def a(n): return nextprime(primorial(n) if n else 1)
print([a(n) for n in range(20)]) # Michael S. Branicky, Dec 24 2022

a(n) = A002110(n) + A038711(n). - Alois P. Heinz, Mar 16 2020

Offset corrected, incorrect comment and formula removed, and more terms added by Jinyuan Wang, Mar 16 2020

A060270 Distance of n-th primorial from previous prime.

Original entry on oeis.org

1, 1, 11, 1, 1, 29, 23, 43, 41, 73, 59, 1, 89, 67, 73, 107, 89, 101, 127, 97, 83, 89, 1, 251, 131, 113, 151, 263, 251, 223, 179, 389, 281, 151, 197, 173, 239, 233, 191, 223, 223, 293, 593, 293, 457, 227, 311, 373, 257, 307, 313, 607, 347, 317, 307, 677, 467, 317

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			Before 7th primorial 510481 is the largest prime. Its distance from 510510 is a(7)=29.

Robert G. Wilson v, Table of n, a(n) for n = 2..1000

Cf. A002110, A038710, A007014, A058044, A038711, A055211.

[seq(product(ithprime(j),j=1..n)-prevprime(product(ithprime(j),j=1..n)), n=2..50)];

Map[# - NextPrime[#, -1] &, Rest@ FoldList[Times, Prime@ Range[59]]] (* Michael De Vlieger, Aug 10 2023 *)

a(n) = my(P=vecprod(primes(n))); P-precprime(P-1); \\ Michel Marcus, Aug 11 2023

a(n)=1 for n=2, 3, 5, 6, 13, 24, 66, 68, 167, ... (A057704); a(n)=A055211(n) otherwise. - Jeppe Stig Nielsen, Oct 31 2003

More terms from Jeppe Stig Nielsen, Oct 31 2003

A058044 Difference between the smallest prime following and largest prime preceding n-th primorial number.

Original entry on oeis.org

2, 2, 12, 2, 18, 48, 46, 80, 102, 74, 120, 72, 136, 174, 132, 168, 198, 190, 230, 176, 234, 286, 102, 354, 364, 336, 278, 486, 442, 386, 408, 1032, 520, 308, 364, 612, 478, 432, 382, 422, 606, 526, 1344, 606, 1230, 834, 624, 756, 550

Offset: 2

Labos Elemer, Nov 17 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..100

Cf. A002110, A007014, A038710.

[seq(nextprime(product(ithprime(k), k=1..w))-prevprime (product(ithprime(k), k=1..w)), w=2..50)];

Rest[NextPrime[#]-NextPrime[#,-1]&/@Rest[FoldList[Times,1,Prime[Range[ 50]]]]] (* Harvey P. Dale, Mar 24 2013 *)

a(n) = A038710(n)-A007014(n).

Offset corrected by Alois P. Heinz, Jun 08 2014

A060269 Distance of n-th primorial from closest prime.

Original entry on oeis.org

1, 1, 1, 1, 19, 23, 37, 41, 1, 59, 1, 47, 67, 59, 61, 89, 89, 103, 79, 83, 89, 1, 103, 131, 113, 127, 223, 191, 163, 179, 389, 239, 151, 167, 173, 239, 199, 191, 199, 223, 233, 593, 293, 457, 227, 311, 373, 257

Offset: 3

Labos Elemer, Mar 23 2001

nonn

			7th primorial is surrounded by {510529,510481} primes in {19,71} distances of which the smaller is 19=a(7).

Cf. A002110, A038710, A007014, A058044, A038711.

with(numtheory): [seq(min(nextprime(product(ithprime(j),j=1..n))-product(ithprime(j),j= >1..n),product(ithprime(j),j=1..n)-prevprime(product(ithprime(j),j=1..n >))), n=3..50)];

dnp[n_]:=Module[{a=NextPrime[n,-1],b=NextPrime[n]},Min[n-a,b-n]]; dnp/@ FoldList[Times,Prime[Range[50]]] (* Harvey P. Dale, Jul 11 2017 *)

A340041 The prime gap, divided by two, which surrounds p#.

Original entry on oeis.org

1, 1, 6, 1, 9, 24, 23, 40, 51, 37, 60, 36, 68, 87, 66, 84, 99, 95, 115, 88, 117, 143, 51, 177, 182, 168, 139, 243, 221, 193, 204, 516, 260, 154, 182, 306, 239, 216, 191, 211, 303, 263, 672, 303, 615, 417, 312, 378, 275, 375, 322, 445, 312, 294, 354, 492, 399, 348, 461

Offset: 2

Robert G. Wilson v, Jan 22 2021

nonn

If p and q are consecutive primes, we say here that there is a gap of q-p. (Other sequences use different definitions of "gap".) - N. J. A. Sloane, Mar 07 2021

Records: 1, 6, 9, 24, 40, 51, 60, 68, 87, 99, 115, 117, 143, 177, 182, 243, 516, 672, 855, 915, 925, 1100, 1139, 1620, 1863, 2272, 2842, 4177, 4190, 5025, 5692, 6254, 6413, 6879, 7914, 8026, 9928, 10604, ..., .

			For a(1), there are two contiguous primes {2, 3} with 2 being 2#. The prime gap is 1. However, the two primes do not surround 2#, so a(1) like A340013(2) is undefined.
For a(2), the prime gap contains {5, 6, 7}, with 3# = 6 in the middle. The prime gap is 2, therefore a(2) = 1;
For a(3), the prime gap contains {29, 30, 31}, with 5# = 30  in the middle. The prime gap is 2, therefore a(3) = 1.
For a(4), the prime gap contains {199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211}, with 7# =  205 in the middle. The prime gap is 12, therefore a(4) = 6. etc.

Cf. A006862, A007014, A038711, A060270, A340013 (analog for n!).

a[n_] := Block[{p = Times @@ Prime@ Range@ n}, (NextPrime[p, 1] - NextPrime[p, -1])/2]; a[1] = 0; Array[a, 60]

a(n) = (A006862(n) - A007014(n))/2 = (A038711(n) + A060270(n))/2.

a(n) = A058044(n)/2. - Hugo Pfoertner, Jan 22 2021

cp's OEIS Frontend

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

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A003604 Number of primes <= n!.

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A038710 a(n) is the smallest prime > product of the first n primes (A002110(n)).

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A060270 Distance of n-th primorial from previous prime.

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A058044 Difference between the smallest prime following and largest prime preceding n-th primorial number.

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A060269 Distance of n-th primorial from closest prime.

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