0,2

See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n.

p(n)# is the least number N with n distinct prime factors (i.e. omega(N)=n, cf. A001221). - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 (blekraj(AT)yahoo.com), 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 (j.mccranie(AT)comcast.net), Jun 11 2005

Comment from David W. Wilson (davidwwilson(AT)comcast.net), Oct 23 2006: 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.

Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.

Records in number of distinct prime divisors - Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008

Carella proves on p. 12 what J.-L. Nicholas asserted in 1983, namely that, if the Riemann Hypothesis is true, a(n)/phi(a(n)) > (e^gamma) log log a(n) for all sufficiently large a(n), where phi is the Euler totient function A000010. Conversely, if the Riemann Hypothesis is false, then both a(n)/phi(a(n)) > (e^gamma) log log a(n) and a(n)/phi(a(n)) < (e^gamma) log log a(n) occur for infinitely many k => 1. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 17 2008. Warning: See the following comments! - N. J. A. Sloane, Jul 21 2009

Comments from Geoffrey Caveney (rokirovka(AT)gmail.com), May 17 2009: (Start)

(Responding to the comment of Jonathan Vos Post about the paper of Carella referenced in the notes.)

The theorem Carella *claims* to prove (his Theorem 7), if true, would

actually amount to a proof of the Riemann Hypothesis when combined with the

theorem of Nicolas (Theorem 6 in Carella's paper):

On page 2 Carella states as Theorem 6 Nicolas' result that (i) if the

Riemann Hypothesis is true, then N_k / phi(N_k) > e^gamma log log(N_k) for

all k >= 1, and (ii) if the Riemann Hypothesis is false, then both N_k /

phi(N_k) < e^gamma log log(N_k) and N_k / phi(N_k) > e^gamma log log(N_k)

occur for infinitely many k >= 1.

Then Carella states as Theorem 7 his own result that N_k / phi(N_k) >

e^gamma log log(N_k) for all sufficiently large integer N_k. He presents his

claimed proof of this result on pages 12-13.

But Carella's paper does not seem to note the fact that if his Theorem 7 is

true and Nicolas' Theorem 6 is true, this amounts to a proof of the Riemann Hypothesis:

If N_k / phi(N_k) > e^gamma log log(N_k) for all sufficiently large integer

N_k, then there can only be finitely many k such that N_k / phi(N_k) <= e^gamma log log(N_k).

Therefore N_k / phi(N_k) < e^gamma log log(N_k) cannot occur for infinitely many k >= 1.

Therefore by Theorem 6-ii, the Riemann Hypothesis cannot be false. Thus the

Riemann Hypothesis is proved to be true.

One would expect to find a flaw in a one-page proof of a result that implies

the Riemann Hypothesis. Here is the first one:

On page 12 Carella begins his proof as follows:

"On the contrary suppose that N_k / phi(N_k) <= e^gamma log log(N_k). Then

log Product_[p|N_k] (1 - 1/p^2)^-1 (1 + 1/p) <= log(e^gamma) log

log(N_k), (8)

see Proposition 8-i."

There is not, however, any Proposition 8-i to be found in his paper. (End)

Successive minimal records in value of EulerPhi[k]/k. [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]

The digital roots of primorial numbers are multiples of 3. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 19 2009]

Denominators of the sum of the ratios of consecutive primes. Cf. A094661 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 24 2009]

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.

S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.

J.-L. Nicholas, Petites valeurs de la fonction d'Euler, J. Number Theory 17(1983)375-388.

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).

Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.

D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.

T. D. Noe, Table of n, a(n) for n = 0..100

C. K. Caldwell, The Prime Glossary, primorial

N. A. Carella, Divisor and Totient Functions Estimates

F. Ellermann, Illustration for A002110, A005867, A038110, A060753

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, Link to a section of The World of Mathematics.

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 (qntmpkt(AT)yahoo.com), Dec 14 2007

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

with (numtheory):a:=n->mul(ithprime(j), j=1..n):seq(a(n), n=0..17); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]

FoldList[Times, 1, Prime[Range[20]]]

max = 0; a = {1}; Do[w = Length[FactorInteger[n]]; If[w > max, AppendTo[a, n]; max = w], {n, 2, 100000}]; a - Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008

aa = {}; min = 2; Do[k = EulerPhi[n]/n; If[k < min, AppendTo[aa, n]; min = k], {n, 1, 200000}]; aa [From Artur Jasinski (grafix(AT)csl.pl), Nov 05 2008]

s=0; lst={}; Do[p=Prime[n]; r=Prime[n+1]; AppendTo[lst, Denominator[s+=r/p]], {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 24 2009]

(PARI) a(n)=prod(i=1, n, prime(i)) - W. Bomfim (webonfim(AT)bol.com.br), Sep 23 2008

(PARI) { p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Nov 13 2009]

Cf. A034387, A005235, A006862, A035345, A035346, A057588, A136104, A121572.

Primorial base representation: A049345.

Squares: A061742.

a(n) = Product[i=1..n] A000040(i). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 17 2008

Cf. A094348, A003418, A002182, A002201, A072938, A106037.

Sequence in context: A129779 A068215 A096775 this_sequence A118491 A088257 A058694

Adjacent sequences: A002107 A002108 A002109 this_sequence A002111 A002112 A002113

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)

1,2

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.

The exponents k1, k2, ... can be read off Abramowitz and Stegun, Handbook, p. 831, column labeled "pi".

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001

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].

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010

The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

PrimeExponents[n_] := Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[n]]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2350}]; ln (from Robert G. Wilson v Aug 14 2004)

Cf. A036035, A025488, A051282. Equals range of values taken by A046523.

Cf. A055932, A036041, A061394, A124832.

Sequence in context: A048951 A058629 A095810 this_sequence A070175 A096850 A062847

Adjacent sequences: A025484 A025485 A025486 this_sequence A025488 A025489 A025490

nonn,easy,nice

David W. Wilson (davidwwilson(AT)comcast.net)

Offset corrected by Matt Vandermast, Oct 19 2008

0,1

lst={}; r=1; Do[p=Prime[n]; r=r*p; q=r-1; If[PrimeQ[q], (*Print[p]; *)AppendTo[lst, q]], {n, 1, 10^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 22 2008]

See A006794 and A057704 (the main entries for this sequence) for more terms. Cf. A014545, A002110.

Sequence in context: A057208 A046842 A057706 this_sequence A086720 A056869 A098346

Adjacent sequences: A057702 A057703 A057704 this_sequence A057706 A057707 A057708

nonn,nice

Labos E. (labos(AT)ana.sote.hu), Oct 24 2000

1,1

Equivalently, smaller of twin prime pair with primes in different decades.

For the pair {149,151} (149+151)/2 = 5*30.

A001359, A002110, A060230, A060231, A158277, A158861.

Sequence in context: A042672 A042670 A129813 this_sequence A139507 A104119 A042674

Adjacent sequences: A060226 A060227 A060228 this_sequence A060230 A060231 A060232

nonn

Labos E. (labos(AT)ana.sote.hu), Mar 21 2001

Minor edits by Ray Chandler (rayjchandler(AT)sbcglobal.net), Apr 02 2009

1,3

Cf. A002110, A057704, A128421.

Sequence in context: A112234 A023983 A159061 this_sequence A099135 A047282 A064437

Adjacent sequences: A128417 A128418 A128419 this_sequence A128421 A128422 A128423

nonn

Pierre CAMI (pierrecami(AT)tele2.fr), Mar 02 2007

Corrected by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2007

Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net), Mar 13 2007

1,1

Cf. A002110, A014545, A128420.

Sequence in context: A119336 A133706 A081710 this_sequence A115885 A029964 A141655

Adjacent sequences: A128418 A128419 A128420 this_sequence A128422 A128423 A128424

nonn

Pierre CAMI (pierrecami(AT)tele2.fr), Mar 02 2007

Corrected by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2007

Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net), Mar 13 2007

0,3

C. D. Pruitt, A Theorem & Proof on the Density of Primes Utilizing Primorials [Broken link?]

a=1; Table[a=a*Prime[n]; PrimePi[a], {n, 1, 13}]

Cf. A000720, A002110, A003604.

Sequence in context: A058112 A020008 A167999 this_sequence A092429 A005651 A105748

Adjacent sequences: A000846 A000847 A000848 this_sequence A000850 A000851 A000852

nonn,hard,more

James D. Ausfahl, gandalf(AT)hrn.office.ssi.net

More terms from David W. Wilson (davidwwilson(AT)comcast.net); last 4 terms from Paul.Zimmermann(AT)loria.fr (Paul Zimmermann).

1,2

These orders determines the maximal numbers of unitary divisors valid for integers in given binary order ranges (see A046971).

The sixth primorial number is Q_6=2*3*5*7*11*13=30030 which is in the range of [ 16384,32766 ]=[ 2^14,2^15 ], so its binary order is a[ 6 ]=15

Cf. A029837, A002110, A046971.

A054850(n)+1.

Sequence in context: A153400 A100464 A114891 this_sequence A122539 A023660 A161339

Adjacent sequences: A045713 A045714 A045715 this_sequence A045717 A045718 A045719

nonn

Labos E. (labos(AT)ana.sote.hu)

1,1

Dickson, L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.

a(n)=A058989(n)+1 (see that entry for much more information).

Cf. A048669, A002110, A005867.

Sequence in context: A023499 A103445 A001747 this_sequence A077625 A027383 A138016

Adjacent sequences: A048667 A048668 A048669 this_sequence A048671 A048672 A048673

nonn,nice

Jan Kristian Haugland (jankrihau(AT)hotmail.com)

4 more terms from Max Alekseyev, Apr 09 2006

Terms a(25) onwards from Tom Hagedorn (hagedorn(AT)tcnj.edu), Feb 21 2007

1,1

First differences of reduced residue systems modulo primorial numbers are essentially palindromic + 1 separator term (2). The palindromic part starts and ends with p_(n+1)-1 for the n-th primorial number.

This sequence has period A005867(4)=A000010(A002110(4))=48. The 0th, first, 2nd and 3rd similar difference sequences are as follows: {1},{2},{4,2},{6,4,2,4,2,4,6,2} obtained from reduced residue systems of consecutive primorials.

Difference sequence of the "4th diatomic sequence" - A. de Polignac (1849), J. Dechamps (1907).

Dickson L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.

t1=Table[ GCD[ w, 210 ], {w, 1, 210} ] /t2=Flatten[ Position[ t1, 1 ] ] /t3=Mod[ RotateLeft[ t2 ]-t2, 210 ]

Cf. A005867, A008364, A002110, A001223.

Sequence in context: A138999 A010175 A160136 this_sequence A161995 A069036 A155817

Adjacent sequences: A049293 A049294 A049295 this_sequence A049297 A049298 A049299

nonn,easy,nice

Labos E. (labos(AT)ana.sote.hu)

Corrected by Frederic Devaux, Feb 02 2007