A249270 -id:A249270 - cp's OEIS frontend

A339306 Continued fraction expansion of the constant defined in A249270.

2, 1, 11, 1, 1, 30, 1, 6, 2, 27, 1, 3, 41, 1, 4, 19, 2, 1, 3, 2, 1, 3, 3, 12, 1, 1, 12, 1, 60, 1, 3, 1, 1, 3, 4, 1, 5, 3, 1, 4, 4, 6, 1, 3, 5, 1, 1, 1, 17, 1, 1, 1, 1, 3, 1, 2, 6, 8, 1, 11, 1, 1, 80, 1, 7, 3, 11, 1, 1, 2, 1, 105, 1, 3, 14, 1, 10, 1, 22, 5, 2

Offset: 0

Michael Reilly, Nov 29 2020

Michael Reilly, Table of n, a(n) for n = 0..10000

Cf. A249270, A034386.

q[x_] := Apply[Times, Table[Prime[w], {w, 1, PrimePi[x]}]];
ContinuedFraction[Sum[1/q[n], {n, 0, 100}]]

A341930 Decimal expansion of (A249270 + A340469)/2.

Original entry on oeis.org

2, 0, 6, 7, 4, 3, 5, 8, 9, 1, 4, 2, 0, 2, 3, 4, 2, 2, 9, 5, 1, 8, 8, 4, 7, 0, 7, 5, 6, 6, 7, 8, 9, 5, 2, 1, 1, 1, 3, 0, 9, 3, 9, 0, 7, 9, 7, 6, 6, 8, 4, 9, 5, 5, 5, 6, 7, 6, 5, 4, 8, 5, 7, 6, 2, 0, 0, 7, 7, 3, 8, 8, 1, 5, 5, 3, 7, 6, 4, 6, 9, 2, 3, 9, 7, 1, 1, 8, 5, 8, 6, 3, 2, 5, 5, 7, 0, 0, 0, 3, 9, 6, 8

Offset: 1

Davide Rotondo, Feb 23 2021

With this constant r(1) and using the formula r(n+1) = round(r(n))*(r(n) - round(r(n)) + 1.5) it is possible to obtain the sequence of prime numbers because round(r(n)) = prime(n).

			2.06743589142023422951884707566789...

Cf. A249270, A340469.

suminf(k=1, (prime(k)-1.5)/prod(i=1, k-1, prime(i))) \\ Michel Marcus, Feb 23 2021

r(1) = Sum_{k>=1} (prime(k)-1.5)/Product_{i=1..k-1} prime(i).

A346801 Engel expansion of A249270 = lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 13, 18, 160, 162, 1421, 1745, 6097, 6348, 15474, 65948, 103608, 366088, 573005, 1048188, 1138953, 3410520, 6376825, 279823002, 306445433, 1082288597, 1489247033, 2533043524, 5591690612, 8082600305, 22159965172, 62442143259, 70275959860

Offset: 1

Corey Clemons, Aug 04 2021

nonn

Cf. A006784 for definition of Engel expansion.

Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A006784, A249270.

More terms from Alois P. Heinz, Aug 04 2021

A034386 Primorial numbers (second definition): n# = product of primes <= n.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230, 200560490130, 200560490130

Offset: 0

N. J. A. Sloane

Squarefree kernel of both n! and lcm(1, 2, 3, ..., n).
a(n) = lcm(core(1), core(2), core(3), ..., core(n)) where core(x) denotes the squarefree part of x, the smallest integer such that x*core(x) is a square. - Benoit Cloitre, May 31 2002
The sequence can also be obtained by taking a(1) = 1 and then multiplying the previous term by n if n is coprime to the previous term a(n-1) and taking a(n) = a(n-1) otherwise. - Amarnath Murthy, Oct 30 2002; corrected by Franklin T. Adams-Watters, Dec 13 2006

			a(5) = a(6) = 2*3*5 = 30;
a(7) = 2*3*5*7 = 210.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, p. 14, "n?".
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.35, p. 268.

Alois P. Heinz, Table of n, a(n) for n = 0..2370 (first 401 terms from T. D. Noe)
Jens Askgaard, On the additive period length of the Sprague-Grundy function of certain Nim-like games, arXiv:1902.06299 [math.CO], 2019.
Klaus Dohmen and Martin Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv:1404.5480 [math.CO], 2014.
Romeo Meštrović, 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. - From _N. J. A. Sloane_, Jun 13 2012
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), 64-94.
Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A057872, A249270.
Cf. A073838, A034387. - Reinhard Zumkeller, Jul 05 2010
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.

[n eq 0 select 1 else LCM(PrimesInInterval(1, n)) : n in [0..50]]; // G. C. Greubel, Jul 21 2023

A034386 := n -> mul(k,k=select(isprime,[$1..n])); # Peter Luschny, Jun 19 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      `if`(isprime(n), n, 1)*a(n-1))
    end:
seq(a(n), n=0..36);  # Alois P. Heinz, Nov 26 2020

q[x_]:=Apply[Times,Table[Prime[w],{w,1,PrimePi[x]}]]; Table[q[w],{w,1,30}]
With[{pr=FoldList[Times,1,Prime[Range[20]]]},Table[pr[[PrimePi[n]+1]],{n,0,40}]] (* Harvey P. Dale, Apr 05 2012 *)
Table[ResourceFunction["Primorial"][i], {i,1,40}] (* Navvye Anand, May 22 2024 *)

a(n)=my(v=primes(primepi(n)));prod(i=1,#v,v[i]) \\ Charles R Greathouse IV, Jun 15 2011

a(n)=lcm(primes([2,n])) \\ Jeppe Stig Nielsen, Mar 10 2019

from sympy import primorial
def A034386(n): return 1 if n == 0 else primorial(n,nth=False) # Chai Wah Wu, Jan 11 2022

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
[sharp_primorial(n) for n in (0..30)] # Giuseppe Coppoletta, Jan 26 2015

a(n) = n# = A002110(A000720(n)) = A007947(A003418(n)) = A007947(A000142(n)).
Asymptotic expression for a(n): exp((1 + o(1)) * n) where o(1) is the "little o" notation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, log(a(n)) < 1.01624*n. [Rosser and Schoenfeld, 1962; Sándor et al., 2005] - N. J. A. Sloane, Apr 04 2017
a(n) <= A179215(n). - Reinhard Zumkeller, Jul 05 2010
a(n) = lcm(A006530(n), a(n-1)). - Jon Maiga, Nov 10 2018
Sum_{n>=0} 1/a(n) = A249270. - Amiram Eldar, Nov 08 2020

Offset changed and initial term added by Arkadiusz Wesolowski, Jun 04 2011

A053669 Smallest prime not dividing n.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2

Offset: 1

Henry Bottomley, Feb 15 2000

Smallest prime coprime to n.
Smallest k >= 2 coprime to n.
a(#(p-1)) = a(A034386(p-1)) = p is the first appearance of prime p in sequence.
a(A005408(n)) = 2; for n > 2: a(n) = A112484(n,1). - Reinhard Zumkeller, Sep 23 2011
Average value is 2.920050977316134... = A249270. - Charles R Greathouse IV, Nov 02 2013
Differs from A236454, "smallest number not dividing n^2", for the first time at n=210, where a(210)=11 while A236454(210)=8. A235921 lists all n for which a(n) differs from A236454. - Antti Karttunen, Jan 26 2014
For k >= 0, a(A002110(k)) is the first occurrence of p = prime(k+1). Thereafter p occurs whenever A007947(n) = A002110(k). Thus every prime appears in this sequence infinitely many times. - David James Sycamore, Dec 04 2024

			a(60) = 7, since all primes smaller than 7 divide 60 but 7 does not.
a(90) = a(120) = a(150) = a(180) = 7 because 90,120,150,180 all have same squarefree kernel = 30 = A002110(3), and 7 is the smallest prime which does not divide 30. - _David James Sycamore_, Dec 04 2024

T. D. Noe, Table of n, a(n) for n = 1..10000
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73; ResearchGate link.
James Grime and Brady Haran, 2.920050977316, Numberphile video (2020)
Igor Rivin, Geodesics with one self-intersection, and other stories, Advances in Mathematics, Vol. 231, No. 5 (2012), pp. 2391-2412; arXiv preprint, arXiv:0901.2543 [math.GT], 2009-2011.

One more than A071222(n-1).
Cf. A000040, A020639, A053670, A053671, A053672, A053673, A053674, A055874, A079578, A087560, A096014, A235921, A236454, A249270, A257993 (the indices of these primes), A276086, A324895, A353526, A353528, A353529, A358754, A358755, A370125.
Cf. A002110, A007947.

a053669 n = head $ dropWhile ((== 0) . (mod n)) a000040_list
-- Reinhard Zumkeller, Nov 11 2012

f:= proc(n) local p;
p:= 2;
while n mod p = 0 do p:= nextprime(p) od:
p
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2016

Table[k := 1; While[Not[GCD[n, Prime[k]] == 1], k++ ]; Prime[k], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
With[{prs=Prime[Range[10]]},Flatten[Table[Select[prs,!Divisible[ n,#]&,1],{n,110}]]] (* Harvey P. Dale, May 03 2012 *)

a(n)=forprime(p=2,,if(n%p,return(p))) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import nextprime
def a(n):
    p = 2
    while True:
        if n%p: return p
        else: p=nextprime(p) # Indranil Ghosh, May 12 2017

# using standard library functions only
import math
def a(n):
    k = 2
    while math.gcd(n,k) > 1: k += 1
    return k # Ely Golden, Nov 26 2020

(define (A053669 n) (let loop ((i 1)) (cond ((zero? (modulo n (A000040 i))) (loop (+ i 1))) (else (A000040 i))))) ;; Antti Karttunen, Jan 26 2014

a(n) = A071222(n-1)+1. [Because the right hand side computes the smallest k >= 2 such that gcd(n,k) = gcd(n-1,k-1) which is equal to the smallest k >= 2 coprime to n] - Antti Karttunen, Jan 26 2014
a(n) = 1 + Sum_{k=1..n}(floor((n^k)/k!)-floor(((n^k)-1)/k!)) = 2 + Sum_{k=1..n} A001223(k)*( floor(n/A002110(k))-floor((n-1)/A002110(k)) ). - Anthony Browne, May 11 2016
a(n!) = A151800(n). - Anthony Browne, May 11 2016
a(2k+1) = 2. - Bernard Schott, Jun 03 2019
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A249270. - Amiram Eldar, Oct 29 2020
a(n) = A000040(A257993(n)) = A020639(A276086(n)) = A276086(n) / A324895(n). - Antti Karttunen, Apr 24 2022
a(n) << log n. For every e > 0, there is some N such that for all n > N, a(n) < (1 + e)*log n. - Charles R Greathouse IV, Dec 03 2022
A007947(n) = A002110(k) ==> a(n) = prime(k+1). - David James Sycamore, Dec 04 2024

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000 and James Sellers, Feb 22 2000

Entry revised by David W. Wilson, Nov 25 2006

A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...

Original entry on oeis.org

7, 0, 5, 2, 3, 0, 1, 7, 1, 7, 9, 1, 8, 0, 0, 9, 6, 5, 1, 4, 7, 4, 3, 1, 6, 8, 2, 8, 8, 8, 2, 4, 8, 5, 1, 3, 7, 4, 3, 5, 7, 7, 6, 3, 9, 1, 0, 9, 1, 5, 4, 3, 2, 8, 1, 9, 2, 2, 6, 7, 9, 1, 3, 8, 1, 3, 9, 1, 9, 7, 8, 1, 1, 4, 8, 0, 0, 2, 8, 6, 3, 5, 8, 6, 1, 1, 9, 0, 5, 1, 9, 8, 4, 0, 2, 7, 4, 7, 6, 6, 5, 9, 2, 5, 6

Offset: 0

Labos Elemer, Oct 04 2001

The Engel expansion of this constant is the sequence of primes. - Jonathan Vos Post, May 04 2005
Let S be the operator over the space omega of infinite sequences of numbers, defined to be the Engel expansion of the sum of reciprocals of primorials of a sequence p of numbers; than the eigenvalue-equation S p = p is satisfied by the sequence of prime numbers. - Ralf Steiner, Dec 31 2016
This constant is irrational (Griffiths, 2015). - Amiram Eldar, Oct 27 2020

			0.705230171791800965147431682888248513743577639109154328192267913813919...

Friedrich Engel, "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Friedrich Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
Martin Griffiths, 99.29 On the sum of the reciprocals of the primorials, The Mathematical Gazette, Vol. 99, No. 546 (2015), pp. 522-523.
Simon Plouffe, Sum of the product of prime reciprocals.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A002110, A054543, A000027, A053977, A006784, A028259, A165509 (continued fraction).

Cf. A249270, A340469.

RealDigits[ Sum[1/Product[ Prime[i], {i, n}], {n, 58}], 10, 111][[1]] (* Robert G. Wilson v, Aug 05 2005 *)
RealDigits[Total[1/#&/@FoldList[Times,Prime[Range[100]]]],10,120][[1]] (* Harvey P. Dale, Aug 27 2019 *)

default(realprecision, 20080); p=1; s=x=0; for (k=1, 10^9, p*=prime(k); s+=1.0/p; if (s==x, break); x=s ); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b064648.txt", n, " ", d)) \\ Harry J. Smith, Sep 21 2009

@CachedFunction
def pv(n):
    a = 1
    b = 0
    for i in (1..n):
        a *= nth_prime(i)
        b += 1/a
    return b
N(pv(100),digits=108) # From Maple code Jani Melik, Jul 22 2015

(1/2)*(1 + (1/3)*(1 + (1/5)*(1 + (1/7)*(1 + (1/11)*(1 + (1/13)*(1 + ...)))))). - Jonathan Sondow, Aug 04 2014

Equals Sum_{n>=1} 1/A002110(n). - Amiram Eldar, Oct 27 2020

A232927 a(n) is the smallest k such that the first k primes generate the multiplicative group modulo n.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 1, 2, 4, 3, 2, 3, 1, 5, 3, 4, 3, 6, 1, 3, 1, 3, 1, 5, 2, 3, 3, 2, 2, 4, 1, 2, 4, 5, 2, 5, 2, 4, 4, 3, 3, 6, 2, 2, 3, 3, 1, 3, 2, 5, 3, 2, 1, 6, 1, 2, 3, 3, 2, 4, 1, 3, 3, 8, 4, 6, 3, 3, 4, 4, 2, 4, 2, 5, 1, 3, 1, 6, 2, 2, 3, 4, 2, 5, 2, 3, 5, 3, 4, 6, 3, 2, 3, 5, 1, 4, 2, 5, 6, 2, 1, 4, 2, 4, 5, 5, 2, 6, 4, 4, 4, 5, 5, 7

Offset: 3

Steven Finch, Dec 02 2013

nonn

H. Brown and H. Zassenhaus, Some empirical observations on primitive roots, J. Number Theory 3 (1971) 306-309.
S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdos, J. Number Theory 132 (2012) 1185-1202.

Cf. A098990, A249270.

A260188 Greatest primorial less than or equal to n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30

Offset: 1

Jean-Marc Rebert, Jul 18 2015

nonn

			a(5) = 2 because 2 is the greatest primorial less than or equal to 5.
a(31) = 30 because 30 is the greatest primorial less than or equal to 31.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A034386 (primorials), A048764, A249270.

Table[k = 0; While[Times @@ Prime@ Range[k + 1] <= n, k++]; Times @@ Prime@ Range@ k, {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)

a(n)=my(t=1,k); forprime(p=2,, k=t*p; if(k>n, return(t), t=k)) \\ Charles R Greathouse IV, Jul 20 2015

a(n) = max_{A034386(i) <= n} A034386(i).
a(n) >> n/log n. - Charles R Greathouse IV, Jul 20 2015
Sum_{n>=1} 1/a(n)^2 = A249270. - Amiram Eldar, Aug 09 2022

A071222 Smallest k such that gcd(n,k) = gcd(n+1,k+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 0

Benoit Cloitre, Jun 10 2002

a(n) = least m>0 such that gcd(n!+1+m,n-m) = 1.  [Clark Kimberling, Jul 21 2012]
From Antti Karttunen, Jan 26 2014: (Start)
a(n-1)+1 = A053669(n) = Smallest k >= 2 coprime to n = Smallest prime not dividing n.
Note that a(n) is equal to A235918(n+1) for the first 209 values of n. The first difference occurs at n=210 and A235921 lists the integers n for which a(n) differs from A235918(n+1).
(End)

Clark Kimberling & Antti Karttunen, Table of n, a(n) for n = 0..10001 (Terms up to n=1000 from Kimberling)

One less than A053669(n+1).

Cf. also A007978, A055874, A235918, A235921, A249270.

a071222 n = head [k | k <- [1..], gcd (n + 1) (k + 1) == gcd n k]
-- Reinhard Zumkeller, Oct 01 2014

sgcd[n_]:=Module[{k=1},While[GCD[n,k]!=GCD[n+1,k+1],k++];k]; Array[sgcd,110] (* Harvey P. Dale, Jul 13 2012 *)

PARI
```
for(n=1,140,s=1; while(gcd(s,n)
				
```

(define (A071222 n) (let loop ((k 1)) (cond ((= (gcd n k) (gcd (+ n 1) (+ k 1))) k) (else (loop (+ 1 k)))))) ;; Antti Karttunen, Jan 26 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A249270 - 1. - Amiram Eldar, Jul 26 2022

Added a(0)=1. - N. J. A. Sloane, Jan 19 2014

A079578 Least number coprime to n and greater than n+1.

Original entry on oeis.org

3, 5, 5, 7, 7, 11, 9, 11, 11, 13, 13, 17, 15, 17, 17, 19, 19, 23, 21, 23, 23, 25, 25, 29, 27, 29, 29, 31, 31, 37, 33, 35, 35, 37, 37, 41, 39, 41, 41, 43, 43, 47, 45, 47, 47, 49, 49, 53, 51, 53, 53, 55, 55, 59, 57, 59, 59, 61, 61, 67, 63, 65, 65, 67, 67, 71, 69, 71, 71, 73, 73

Offset: 1

Reinhard Zumkeller, Jan 24 2003

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

Cf. A053669, A116933, A116934, A249270.

a079578 n = head [m | m <- [n + 2 ..], gcd m n == 1]
-- Reinhard Zumkeller, Oct 01 2014

a[n_] := Module[{p = 2}, While[Divisible[n, p], p = NextPrime[p]]; n + p]; Array[a, 100] (* Amiram Eldar, Apr 13 2025 *)

a(n) = n + A053669(n).
a(n) = (A116934(n) - n)/A116933(n). - Reinhard Zumkeller, Feb 27 2006
Sum_{k=1..n} a(k) ~ n^2 / 2 + c * n, where c = A249270. - Amiram Eldar, Apr 13 2025

cp's OEIS Frontend

A339306 Continued fraction expansion of the constant defined in A249270.

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A341930 Decimal expansion of (A249270 + A340469)/2.

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A346801 Engel expansion of A249270 = lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

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A034386 Primorial numbers (second definition): n# = product of primes <= n.

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A053669 Smallest prime not dividing n.

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A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...

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A232927 a(n) is the smallest k such that the first k primes generate the multiplicative group modulo n.

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