A006862 -id:A006862 - cp's OEIS frontend

A014545 Numbers k such that the k-th Euclid number A006862(k) = 1 + (Product of first k primes) is prime.

0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, 637491

Offset: 1

Eric W. Weisstein, Murray R. Bremner

			a(1) = 0 because the (empty) product of 0 primes is 1, plus 1 yields the prime 2.
prime(4413) = 42209 and Primorial(4413) + 1 = 42209# + 1 is a 18241-digit prime.
prime(13494) = 145823 and Primorial(13494) + 1 = 145823# + 1 is a 63142-digit prime.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

C. K. Caldwell, Prime Pages: Database Search
C. K. Caldwell, Primorial Primes.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
Eric Weisstein's World of Mathematics, Euclid Number
Eric Weisstein's World of Mathematics, Primorial Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A005234 (values of p such that 1 + product of primes <= p is prime).
Cf. A018239 (primorial plus 1 primes).
Cf. A002110, A006862, A057704.

P:= 1:
p:= 1:
count:= 0:
for n from 1 to 1000 do
  p:= nextprime(p);
  P:= P*p;
  if isprime(P+1) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i], i=1..count); # Robert Israel, Nov 04 2015

Flatten[Position[Rest[FoldList[Times,1,Prime[Range[180]]]]+1,?PrimeQ]] (* _Harvey P. Dale, May 04 2012 *) (* this program generates the first 9 positive terms of the sequence; changing the Range constant to 33237 will generate all 23 terms above, but it will take a long time to do so *)

is(n)=ispseudoprime(prod(i=1,n,prime(i))+1) \\ Charles R Greathouse IV, Mar 21 2013

P=1; n=0; forprime(p=1, 10^5, if(ispseudoprime(P+1), print1(n", ")); n=n+1; P*=p;) \\ Hans Loeblich, May 10 2019

a(n+1) = A000720(A005234(n)). - M. F. Hasler, May 31 2018

More terms from Labos Elemer
a(21) from Arlin Anderson (starship1(AT)gmail.com), Oct 20 2000
a(22)-a(23) from Eric W. Weisstein, Mar 13 2004 (based on information in A057704)
Offset and first term changed by Altug Alkan, Nov 27 2015
a(24) from Jeppe Stig Nielsen, Aug 08 2024
a(25) from Jeppe Stig Nielsen, Sep 01 2024
a(26) from Jeppe Stig Nielsen, Sep 24 2024
a(27) from Jeppe Stig Nielsen, Nov 10 2024
a(28) from Jeppe Stig Nielsen, Aug 21 2025

A171989 a(n) = A000010(A006862(n)).

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 29464, 476928, 9671392, 222388792, 6438663000, 200560490130, 7379606916000, 299261862900000, 13004421443456272, 614231422273479360, 31727029501157817600, 1915248189055217892480, 116762424492324428512272

Offset: 0

Giovanni Teofilatto, Jan 21 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 0..98

Cf. A000010, A006862.

A006862 := proc(n) if n = 0 then 2; else 1+mul(ithprime(j),j=1..n) ; end if: end proc: A171989 := proc(n) numtheory[phi](A006862(n)) ; end proc: seq(A171989(n),n=1..18) ; # R. J. Mathar, Jan 30 2010

a(n)=eulerphi(prod(i=1,n,prime(i))+1) \\ Charles R Greathouse IV, Mar 05 2013

a(1) inserted and extended beyond a(5) by R. J. Mathar, Jan 30 2010

Offset changed to 0 and a(0) prepended by Amiram Eldar, Nov 30 2024

A369246 Irregular triangle read by rows, where row n lists in ascending order all numbers k in A046316 for which k' = the n-th Euclid number, where k' stands for the arithmetic derivative, and the Euclid numbers are given by A006862. Rows of length zero are simply omitted, i.e., when A369245(n) = 0.

Original entry on oeis.org

399, 4809, 5763, 63021, 76449, 1301673, 19204051701, 421177029231, 908999759928891, 39248269334566041, 39248273246018313, 68437232802099093891, 4903038892893242229501

Offset: 1

Antti Karttunen, Jan 22 2024

All terms are multiples of 3 because for all n >= 2, A006862(n) = 1 + A002110(n) == +1 (mod 3), see A369252.

Although the first thirteen terms appear in the ascending order, this might not be true for all the later terms, if they exist.

			Rows 1..3 have no terms.
Row 4 has one term: 399 = 3 * 7 * 19, whose arithmetic derivative (see A003415) 399' is 211 = 1 + prime(4)# [= A006862(4)].
Row 5 has two terms: 4809 = 3 * 7 * 229 and 5763 = 3 * 17 * 113, with 4809' = 5763' = 2311 = 1 + prime(5)#.
Row 6 has two terms: 63021 = 3 * 7 * 3001 and 76449 = 3 * 17 * 1499, with 63021' = 76449' = 30031 = 1 + prime(6)#.
Row 7 has one term: 1301673 = 3 * 17 * 25523, whose arithmetic derivative is 510511 = 1 + prime(7)#.
Rows 8 and 9 have no terms.
Row 10 has one term: 19204051701 = 3 * 281 * 22780607, whose arithmetic derivative is 6469693231 = 1 + prime(10)#.
Row 11 has one term: 421177029231 = 3 * 7 * 20056049011, whose arithmetic derivative is 200560490131 = 1 + prime(11)#.
Row 12 has no terms.
Row 13 has one term: 908999759928891 = 3 * 727 * 416781182911, whose arithmetic derivative is 304250263527211 = 1 + prime(13)#.
Row 14 has two terms: 39248269334566041 = 3 * 8071457 * 1620866771, and 39248273246018313 = 3 * 11056387 * 1183276033, which both have arithmetic derivative 13082761331670031 = 1 + prime(14)#.
Row 15 has no terms.
Row 16 has one term: 68437232802099093891 = 3 * 7 * 3258915847719004471, whose arithmetic derivative is 32589158477190044731 = 1 + prime(16)#.
Row 17 has at least this term: 4903038892893242229501 = 3 * 17 * 96138017507710631951, whose arithmetic derivative is 1922760350154212639071 = 1 + prime(17)#.

Subsequence of A008585 and of A046316.
Cf. A003415, A002110, A006862, A369054, A369245 (counts of solutions, length of row n), A369252.
Cf. also A366890, A369240 for similar tables.

A376416 a(n) = A276085(A006862(n)), where A276085 is the primorial base log-function, and A006862 is the Euclid numbers, one more than primorials.

Original entry on oeis.org

1, 2, 30, 6469693230, 7799922041683461553249199106329813876687996789903550945093032474868511536164700810

Offset: 0

Antti Karttunen, Nov 17 2024

nonn

Numbers k such that when we apply primorial base exp function (A276086) twice to them, the results are squarefree even semiprimes, A100484 after its initial 4. See comments in A377871.
a(5)..a(8) have 976, 209, 111, 12051 decimal digits.
a(n) is a primorial for those n that are in A014545, that is, when A006862(n) is one of the primorial primes, A018239.

Cf. A002110, A006862, A014545, A018239, A100484, A276085, A276086, A276087, A377871.

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); };
A376416(n) = A276085(1+prod(i=1,n,prime(i)));

a(n) = A276085(1+A002110(n)) = A276085(A276085(A100484(1+n))).

For n >= 1, A276087(a(n)) = A100484(1+n).

A281318 Number of consecutive nonprime numbers following Euclid numbers A006862.

Original entry on oeis.org

1, 3, 5, 11, 21, 15, 17, 21, 35, 59, 65, 59, 69, 45, 105, 57, 59, 107, 87, 101, 77, 149, 195, 99, 101, 231, 221, 125, 221, 189, 161, 227, 641, 237, 155, 165, 437, 237, 197, 189, 197, 381, 231, 749, 311, 771, 605, 311, 381, 291, 441, 329, 281, 275, 269, 399

Offset: 1

Olivier Bélot, Jan 20 2017

nonn

For n > 1, a(n) >= prime(n), with equality if and only if A006862(n) + prime(n) + 1 is prime. Equality occurs for n=2, 3, 7, 17. Are there any others? - Robert Israel, Jan 30 2017

			a(3) = 5 because primorial p_3# = 5# = 2*3*5 = 30 thus 31 is the third Euclid number, and there are 5 consecutive nonprime numbers {32,33,34,35,36} between 31 and the next prime, 37. - _Michael De Vlieger_, Jan 20 2017

Robert Israel, Table of n, a(n) for n = 1..469

Cf. A006862, A002110.

p:= 0: pn:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
pn:= pn*p;
A[n]:= nextprime(pn+1)-(pn+2);
od:
seq(A[n],n=1..100); # Robert Israel, Jan 30 2017

Table[Function[p, NextPrime@ p - p - 1][Times @@ Prime@ Range@ n + 1], {n, 56}] (* Michael De Vlieger, Jan 20 2017 *)

NextPrime[pn# + 1] - pn# - 1

More terms from Michael De Vlieger, Jan 20 2017

A216205 Incidences of n such that A006862(n) - n! is prime where A006862 are the Euclid numbers.

Original entry on oeis.org

0, 1, 2, 6, 8, 19, 94, 226, 2277, 2742, 2868

Offset: 0

Frank M Jackson, Mar 12 2013

			a(3) = 6 because A006862(6) - 6! = 30031-720 = 29311 and is the 3rd such prime.

Cf. A006862, A014545.

primeproduct[q_] := Product[Prime[r], {r, 1, q}]; nextterm[n_] := (p=n+1; While[!PrimeQ[primeproduct[p]+1-p!], p++]; p); Table[Nest[nextterm, 0, m], {m, 1, 5}] (* changing 5 to 10 will give all 10 terms but takes a long time *)

A261558 Euclid numbers (A006862) of the form 3(ii + ij + jj + i + j) + 1 where i and j are integers.

Original entry on oeis.org

7, 31, 211, 2311, 510511, 6469693231, 200560490131, 304250263527211, 117288381359406970983271, 7858321551080267055879091, 40729680599249024150621323471, 232862364358497360900063316880507363071, 279734996817854936178276161872067809674997231

Offset: 1

Altug Alkan, Nov 18 2015

nonn

Intersection of A006862 and A202822.

			a(1) = 7 because 7 = 2*3 + 1 = 3*(1^2 + 1*0 + 0^2 + 1 + 0) + 1.

Eric Weisstein's World of Mathematics, Euclid Number

Cf. A003136, A006862, A202822.

a(n) = prod(k=1, n, prime(k)) + 1;
isA(n) = if( n<1 || (n%3 == 0), 0, 0 != sumdiv( n, d, kronecker( -3, d)));
for(n=0, 30, if(isA(a(n)), print1(a(n), ", ")))

A267756 Indices of Euclid numbers (A006862) of the form x^2 + y^2 + z^2 where x, y and z are integers.

Original entry on oeis.org

0, 1, 4, 8, 11, 12, 13, 15, 16, 19, 22, 27, 31, 34, 35, 38, 41, 42, 46, 48, 52, 53, 56, 57, 61, 62, 64, 65, 66, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 83, 84, 86, 87, 88, 89, 91, 93, 95, 99, 100, 103, 104, 107, 108, 111, 112, 113, 115, 116, 118, 119, 124, 128, 131, 133

Offset: 1

Altug Alkan, Jan 20 2016

nonn

Corresponding Euclid numbers are 2, 3, 211, 9699691, 200560490131, 7420738134811, 304250263527211, 614889782588491411, 32589158477190044731, ...
Complement of this sequence is 2, 3, 5, 6, 7, 9, 10, 14, 17, 18, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 36, 37, 39, 40, 43, 44, 45, 47, 49, 50, 51, 54, 55, 58, 59, 60, 63, 67, 68, 72, 75, 81, 82, 85, 90, 92, 94, 96, 97, 98, 101, ...
Euclid numbers that are not of the form x^2 + y^2 + z^2 are 7, 31, 2311, 30031, 510511, 223092871, 6469693231, 13082761331670031, 1922760350154212639071, ...

			0 is a term because A006862(0) = 2 = 0^2 + 1^2 + 1^2.
1 is a term because A006862(1) = 3 = 1^2 + 1^2 + 1^2.
4 is a term because A006862(4) = 211 = 3^2 + 9^2 + 11^2.
8 is a term because A006862(8) = 9699691 = 79^2 + 123^2 + 3111^2.

Cf. A004215, A006862.

isA004215(n) = { local(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; }
a006862(n) = prod(k=1, n, prime(k))+1;
for(n=0, 200, if(!isA004215(a006862(n)), print1(n, ", ")));

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

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

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

N. J. A. Sloane

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020

A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024

from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023

;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

More terms from James Sellers, Apr 22 2000

cp's OEIS Frontend

A014545 Numbers k such that the k-th Euclid number A006862(k) = 1 + (Product of first k primes) is prime.

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A171989 a(n) = A000010(A006862(n)).

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A376416 a(n) = A276085(A006862(n)), where A276085 is the primorial base log-function, and A006862 is the Euclid numbers, one more than primorials.

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A281318 Number of consecutive nonprime numbers following Euclid numbers A006862.

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A216205 Incidences of n such that A006862(n) - n! is prime where A006862 are the Euclid numbers.

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A261558 Euclid numbers (A006862) of the form 3*(i*i + i*j + j*j + i + j) + 1 where i and j are integers.

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A267756 Indices of Euclid numbers (A006862) of the form x^2 + y^2 + z^2 where x, y and z are integers.

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A261558 Euclid numbers (A006862) of the form 3(ii + ij + jj + i + j) + 1 where i and j are integers.