A038111 -id:A038111 - cp's OEIS frontend

A161527 Numerators of cumulative sums of rational sequence A038110(k)/A038111(k).

1, 2, 11, 27, 61, 809, 13945, 268027, 565447, 2358365, 73551683, 2734683311, 112599773191, 4860900544813, 9968041656757, 40762420985117, 83151858555707, 5085105491885327, 341472595155548909, 24295409051193284539, 1777124696397561611347

Offset: 1

Daniel Tisdale, Jun 12 2009

By rewriting the sequence of sums as 1 - Product_{n>=1} (1 - 1/prime(n)), one can show that the product goes to zero and the sequence of sums converges to 1. This is interesting because the terms approach 1/(2*prime(n)) for large n, and a sum of such terms might be expected to diverge, since Sum_{n>=1} 1/(2*prime(n)) diverges.
Denominators appear to be given by A060753(n+1). - Peter Kagey, Jun 08 2019
A254196 appears to be a duplicate of this sequence. - Michel Marcus, Aug 05 2019

Peter Kagey, Table of n, a(n) for n = 1..400

Cf. A038110, A038111, A060753, A254196.

Numerator[Table[1 - Product[1 - (1/Prime[k]), {k,1,n}], {n,1,20}]]

r(n) = prod(k=1, n-1, (1 - 1/prime(k)))/prime(n);
a(n) = numerator(sum(k=1, n, r(k))); \\ Michel Marcus, Jun 08 2019

a(n) = A053144(n)/A058250(n). - Jamie Morken, Aug 28 2022

A307388 Length of the period of decimal representation of Product_{k=1..n} A038111(k)/A038110(k).

Original entry on oeis.org

1, 27, 729, 59049, 43046721, 31381059609, 68630377364883, 150094635296999121, 328256967394537077627, 717897987691852588770249, 4710128697246244834921603689, 92709463147897837085761925410587, 3649600726280146254718103955713167842

Offset: 9

Jamie Morken, Apr 06 2019

The offset is 9 since for 0 < n < 5, the product is an integer, and for 4 < n < 9 the decimal expansion ends with zeros.

			For example for n=9 with (2/1) * (6/1) * (15/1) * (105/4) * (385/8) * (1001/16) * (17017/192) * (323323/3072) * (7436429/55296) = 2759414170256180364552625 / 154618822656 = 17846560482454.30745852273604315188195970323350694444444444444... so a(9) = 1.

Jamie Morken, Mathematica Stack Exchange question

Cf. A038111, A038110, A051626, A058250.

Primorial[n_] := Times @@ Prime[Range[n]]
ClearAll[iter]
ClearAll[fracPer, vp];
(*p-adic order*)
vp[p_?PrimeQ, n_Integer] :=
  Length@NestWhileList[#/p &, n/p, IntegerQ] - 1;
(*fraction decimal expansion period*)
fracPer[q_Integer] := 0;
fracPer[q_Rational] := Module[{den, p2, p5}, den = Denominator[q];
   p2 = vp[2, den];
   p5 = vp[5, den];
   den = den/2^p2/5^p5;
   If[den == 1, 0, MultiplicativeOrder[10, den]]];
iter[{periods_, frac_, n_}] := {{periods, fracPer[#]}, #, n + 1} &[
   frac*Primorial[n]/EulerPhi[Primorial[Max[1, n - 1]]]];
Flatten@First@
  Nest[iter, {0, Primorial[0]/EulerPhi[Primorial[0]], 0}, 50]

A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).

Original entry on oeis.org

1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000

Offset: 0

N. J. A. Sloane

Local minima of Euler's phi function. - Walter Nissen
Number of potential primes in a modulus primorial(n+1) sieve. - Robert G. Wilson v, Nov 20 2000
Let p=prime(n) and let p# be the primorial (A002110), then it can be shown that any p# consecutive numbers have exactly a(n-1) numbers whose lowest prime factor is p. For a proof, see the "Proofs Regarding Primorial Patterns" link. For example, if we let p=7 and consider the interval [101,310] containing 210 numbers, we find the 8 numbers 119, 133, 161, 203, 217, 259, 287, 301. - Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 16 2006
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 1, 2, 8, 48, ...) dot (-1, 2, -3, 5, -7, 11, ...).
a(6) = 480 = (1, 1, 1, 2, 8, 48) dot (-1, 2, -3, 5, -7, 11) = (-1, 2, -3, 10, -56, 528). (End)
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
First column of A096294. - Eric Desbiaux, Jun 20 2013
Conjecture: The g.f. for the prime(n+1)-rough numbers (A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063) is x*P(x)/(1-x-x^a(n)+x^(a(n)+1)), where P(x) is an order a(n) polynomial with symmetric coefficients (i.e., c(0)=c(n), c(1)=c(n-1), ...). - Benedict W. J. Irwin, Mar 18 2016
a(n)/A002110(n+1) (primorial(n+1)) is the ratio of natural numbers whose smallest prime factor is prime(n+1); i.e., prime(n+1) coprime to A002110(n). So the ratio of even numbers to natural numbers = 1/2; odd multiples of 3 = 1/6; multiples of 5 coprime to 6 (A084967) = 2/30 = 1/15; multiples of 7 coprime to 30 (A084968) = 8/210 = 4/105; etc. - Bob Selcoe, Aug 11 2016
The 2-adic valuation of a(n) is A057773(n), being sum of the 2-adic valuations of the product terms here. - Kevin Ryde, Jan 03 2023
For n > 1, a(n) is the number of prime(n+1)-rough numbers in [1, primorial(prime(n))]. - Alexandre Herrera, Aug 29 2023

			a(3): the mod 30 prime remainder set sieve representation yields the remainder set: {1, 7, 11, 13, 17, 19, 23, 29}, 8 elements.

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

T. D. Noe, Table of n, a(n) for n = 0..99
Larry Deering, The Black Key Sieve, Box 275, Bellport NY 11713-0275, 1998.
Alphonse de Polignac, Six propositions arithmologiques déduites du crible d'Ératosthène, Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale, Série 1, Tome 8 (1849), pp. 423-429. See p. 425.
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
Ken Hicks and Kevin Ward, Series and Product Relations Made from Primes, arXiv:2108.03268 [math.NT], 2021.
Dennis Martin, Proofs Regarding Primorial Patterns [via Internet Archive Wayback-machine]
Dennis Martin, Proofs Regarding Primorial Patterns [Cached copy, with permission of the author]
Francis E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991.
Travis Near, Improving MATLAB's isprime performance without arbitrary-precision arithmetic, arXiv:2108.04791 [cs.MS], 2021.
John K. Sellers, Distribution of twin primes in repeating sequences of prime factors, arXiv:2108.00288 [math.GM], 2021. See Table 1 p. 11.
Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.

Cf. A000010, A002110, A006093, A054640, A058254, A055768, A070918, A101301, A058251, A060753.
Cf. A057773 (2-adic valuation).
Column 1 of A281890.
Cf. A016035, A345974.

a005867 n = a005867_list !! n
a005867_list = scanl (*) 1 a006093_list
-- Reinhard Zumkeller, May 01 2013

A005867 := proc(n)
    mul(ithprime(j)-1,j=1..n) ;
end proc: # Zerinvary Lajos, Aug 24 2008, R. J. Mathar, May 03 2017

Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
RecurrenceTable[{a[0]==1,a[n]==(Prime[n]-1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Dec 09 2013 *)
EulerPhi@ FoldList[Times, 1, Prime@ Range@ 18] (* Michael De Vlieger, Mar 18 2016 *)

for(n=0, 22, print1(prod(k=1,n, prime(k)-1), ", "))

a(n) = phi(product of first n primes) = A000010(A002110(n)).
a(n) = Product_{k=1..n} (prime(k)-1) = Product_{k=1..n} A006093(n).
Sum_{n>=0} a(n)/A002110(n+1) = 1. - Bob Selcoe, Jan 09 2015
a(n) = A002110(n)-((1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1)). - Jamie Morken, Mar 27 2019
a(n) = |Sum_{k=0..n} A070918(n,k)|. - Alois P. Heinz, Aug 18 2019
a(n) = A058251(n)/A060753(n+1). - Jamie Morken, Apr 25 2022
a(n) = A002110(n) - A016035(A002110(n)) - 1 for n >= 1. - David James Sycamore, Sep 07 2024
Sum_{n>=0} 1/a(n) = A345974. - Amiram Eldar, Jun 26 2025

Offset changed to 0, Name changed, and Comments and Examples sections edited by T. D. Noe, Apr 04 2010

A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 4, 11, 1, 12, 2, 13, 5, 14, 1, 15, 1, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 2, 25, 9, 26, 1, 27, 4, 28, 10, 29, 1, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 1, 37, 13, 38, 3, 39, 1, 40, 14, 41, 1, 42, 6, 43

Offset: 0

Reinhard Zumkeller, Dec 12 2002

nonn

From Antti Karttunen, Dec 06 2014: (Start)
For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.
(End)

Harvey P. Dale (terms 1 - 1000) & Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002110, A008683, A008836, A020639, A032742, A054272, A055396, A078899, A078896, A083140, A083221, A243055, A246277, A249738, A249744, A249808, A249809, A249810, A249820, A249818, A250470, A250474, A250477, A250478, A251719, A251724, A251728.

Cf. A001248, A006094, A038110, A038111, A090076.

import Data.IntMap (empty, findWithDefault, insert)
a078898 n = a078898_list !! n
a078898_list = 0 : 1 : f empty 2 where
   f m x = y : f (insert p y m) (x + 1) where
           y = findWithDefault 0 p m + 1
           p = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

N:= 1000: # to get a(0) to a(N)
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
A:= Vector(N):
for p in Primes do
  t:= 1:
  A[p]:= 1:
  for n from p^2 to N by p do
    if A[n] = 0 then
       t:= t+1:
       A[n]:= t
    fi
  od
od:
0,1,seq(A[i],i=2..N); # Robert Israel, Jan 04 2015

Module[{nn=90,spfs},spfs=Table[FactorInteger[n][[1,1]],{n,nn}];Table[ Count[ Take[spfs,i],spfs[[i]]],{i,nn}]] (* Harvey P. Dale, Sep 01 2014 *)

\\ Not practical for computing, but demonstrates the sum moebius formula:
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A055396(n) = { if(1==n,0,primepi(A020639(n))); };
A002110(n) = prod(i=1, n, prime(i));
A078898(n) = { my(k,p); if(1==n, n, k = A002110(A055396(n)-1); p = A020639(n); sumdiv(k, d, moebius(d)*(n\(p*d)))); };
\\ Antti Karttunen, Dec 05 2014

;; With memoizing definec-macro.
(definec (A078898 n) (if (< n 2) n (+ 1 (A078898 (A249744 n)))))
;; Much better for computing. Needs also code from A249738 and A249744. - Antti Karttunen, Dec 06 2014

Ordinal transform of A020639 (Lpf). - Franklin T. Adams-Watters, Aug 28 2006
From Antti Karttunen, Dec 05-08 2014: (Start)
a(0) = 0, a(1) = 1, a(n) = 1 + a(A249744(n)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(n / (A020639(n)*d)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(A032742(n) / d).
[Instead of Moebius mu (A008683) one could use Liouville's lambda (A008836) in the above formulas, because all primorials (A002110) are squarefree. A020639(n) gives the smallest prime dividing n, and A055396 gives its index].
a(0) = 0, a(1) = 1, a(2n) = n, a(2n+1) = a(A250470(2n+1)). [After a similar recursive formula for A246277. However, this cannot be used for computing the sequence, unless a definition for A250470(n) is found which doesn't require computing the value of A078898(n).]
For n > 1: a(n) = A249810(n) - A249820(n).
(End)
Other identities:
a(2*n) = n.
For n > 1: a(n)=1 if and only if n is prime.
For n > 1: a(n) = A249808(n, A055396(n)) = A249809(n, A055396(n)).
For n > 1: a(n) = A246277(A249818(n)).
From Antti Karttunen, Jan 04 2015: (Start)
a(n) = 2 if and only if n is a square of a prime.
For all n >= 1: a(A251728(n)) = A243055(A251728(n)) + 2. That is, if n is a semiprime of the form prime(i)*prime(j), prime(i) <= prime(j) < prime(i)^2, then a(n) = (j-i)+2.
(End)
a(A000040(n)^2) = 2; a(A000040(n)*A000040(n+1)) = 3. - Reinhard Zumkeller, Apr 06 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Oct 26 2024

a(0) = 0 prepended for recurrence's sake by Antti Karttunen, Dec 06 2014

A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 21, 10, 13, 121, 77, 55, 27, 12, 17, 169, 143, 91, 65, 33, 14, 19, 289, 221, 187, 119, 85, 39, 16, 23, 361, 323, 247, 209, 133, 95, 45, 18, 29, 529, 437, 391, 299, 253, 161, 115, 51, 20, 31, 841, 667, 551, 493, 377, 319, 203, 125, 57, 22

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

A permutation of natural numbers >= 2.
The proportion of integers in the n-th row of the array is given by A005867(n-1)/A002110(n) = A038110(n)/A038111(n). - Peter Kagey, Jun 03 2019, based on comments by Jamie Morken and discussion with Tom Hanlon.
The proportion of the integers after the n-th row of the array is given by A005867(n)/A002110(n). - Tom Hanlon, Jun 08 2019

			Array begins:
   2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})
   3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)
   5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)
   7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)
  11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)
  13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)

Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.
Arrays of integers grouped into rows by various criteria:
by greatest prime factor: A125624,
by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),
by number of distinct prime factors: A125666,
by number of prime factors counted with multiplicity: A078840,
by prime signature: A095904,
by ordered prime signature: A096153,
by number of divisors: A119586,
by number of 1's in binary expansion: A066884 (upward), A067576 (downward),
by distance to next prime: A192179.
Cf. A002110, A005867, A038110, A038111.

a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]
(* second program: *)
rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* Jean-François Alcover, Mar 08 2016 *)

More terms from Hugo Pfoertner and Robert G. Wilson v, Jun 13 2003

A038110 Numerator of frequency of integers with smallest divisor prime(n).

Original entry on oeis.org

1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000

Offset: 1

Wouter Meeussen

Numerator of Product_{k=1..n-1} (1 - 1/prime(k)). - Jonathan Sondow, Jan 31 2014
Equivalently, denominator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A060753). - N. J. A. Sloane, Apr 17 2015
Sum_{n>=1} a(n)/A038111(n) = 1. - Bob Selcoe, Jan 09 2015
a(n)/A038111(n) = (1/prime(n))*Product_{k=1..n-1} (1 - 1/prime(k)) ~ e^(-c)/ (prime(n)*log(prime(n))), where c=0.577... is the Euler constant. - Vladimir Shevelev, Jan 10 2015

			a(10) = 110592 = ( 1*2*4*6*10*12*16*18*22 ) / ( 2*3*5*11 ).

Robert Israel, Table of n, a(n) for n = 1..278
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753
Fred Kline and Gerry Myerson, Identity for frequency of integers with smallest prime(n) divisor, Mathematics Stack Exchange question
Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Int'l J. Math. and Math. Sci. (2008) Art. ID 908045, 1-12. See Eq. (5.8).
Jonathan Sondow and Eric Weisstein, Euler Product, World of Mathematics
Wikipedia, Mertens' theorems

Cf. A038111, A002110, A005867, A058250, A060753, A236435, A236436, A254196.

N:= 100: # for a(1) to a(N)
Q:= 1: p:= 1:
for n from 1 to N do
  p:= nextprime(p);
  A[n]:= numer(Q);
  Q:= Q * (1 - 1/p);
end:
seq(A[n],n=1..N); # Robert Israel, Jul 14 2014

Numerator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 64} ]
(* Wouter Meeussen *)
Numerator@Table[ Product[ 1 - 1/Prime[ k ], {k, n-1}], {n, 64} ]
(* Jonathan Sondow, Jan 31 2014 *)
Numerator@
Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/
Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 21}]
(* Fred Daniel Kline, Jul 14 2014 *)

a(n) = numerator(prod(k=1, n-1, (1 - 1/prime(k)))); \\ Michel Marcus, Aug 05 2019

a(n) = A005867(n-1) / A058250(n-1), where A058250(m) = gcd(A005867(m), A002110(m)). [Edited by Peter Munn, Jun 29 2025]
a(n)/A060753(n) = Product_{k=1..n-1} (1 - 1/prime(k)) ~ exp(-gamma)/log(n) as n->infinity (Mertens's 3rd theorem). - Jonathan Sondow, Jan 31 2014
a(n+1)/A038111(n+1) = a(n)/A038111(n) * (prime(n)-1)/prime(n+1). - Robert Israel, Jul 14 2014
a(n) = numerator of phi(e^(psi(p_n-1)))/e^(psi(p_n)), where psi(.) is the second Chebyshev function and phi(.) is Euler's totient function. - Fred Daniel Kline, Jul 17 2014

A084967 Multiples of 5 whose GCD with 6 is 1.

Original entry on oeis.org

5, 25, 35, 55, 65, 85, 95, 115, 125, 145, 155, 175, 185, 205, 215, 235, 245, 265, 275, 295, 305, 325, 335, 355, 365, 385, 395, 415, 425, 445, 455, 475, 485, 505, 515, 535, 545, 565, 575, 595, 605, 625, 635, 655, 665, 685, 695, 715, 725, 745, 755, 775, 785

Offset: 1

Robert G. Wilson v, Jun 15 2003

Third row of A083140.

Positions of 5 in A020639. - Zak Seidov, Apr 29 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A038110, A038111, A083140, A007310 (5-rough numbers), A273669.
Cf. A020639. - Zak Seidov, Apr 29 2015
Essentially the same as A063149.

5Select[ Range[160], GCD[ #, 2*3] == 1 & ]
Select[Range[5, 785, 10], Mod[#, 3] > 0 &] (* Zak Seidov, Apr 29 2015 *)
a[1] = 5; a[n_] := a[n] = a[n - 1] + 10*(2 - Mod[n, 2]); Table[a[n], {n, 50}] (* Zak Seidov, Apr 29 2015 *)

is(n)=n%5==0 && gcd(n,6)==1 \\ Charles R Greathouse IV, Nov 19 2014

list(lim)=5*select(k->gcd(n,6)==1, [1..lim\5]) \\ Charles R Greathouse IV, Nov 19 2014

Numbers of the form 5k for which gcd(5k, 6) = 1.
a(n) = 5*A007310(n). - Adriano Caroli, Oct 03 2010
From Colin Barker, Feb 24 2013: (Start)
a(n) = 5*(-3 + (-1)^n + 6*n)/2.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: 5*x*(x^2+4*x+1) / ((x-1)^2*(x+1)). (End)
Limit_{n->infinity} a(n)/n = A038111(3)/A038110(3) = 15. - Vladimir Shevelev, Jan 20 2015
For n > 2, a(n) = a(n-2) + 30. - Zak Seidov, Apr 29 2015
a(n) = A007310(A273669(n)). - Antti Karttunen, May 20 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(10*sqrt(3)). - Amiram Eldar, Nov 03 2022

A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).

Original entry on oeis.org

1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909

Offset: 1

Frank Ellermann, Apr 23 2001

Equivalently, numerator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A038110). - N. J. A. Sloane, Apr 17 2015
a(n)/A038110(n) is the supremum of the abundancy index sigma(k)/k = A000203(k)/k of the prime(n-1)-smooth numbers, for n>1 (Laatsch, 1986). - Amiram Eldar, Oct 26 2021
From Amiram Eldar, Jul 10 2022: (Start)
a(n)/A038110(n) is the sum of the reciprocals of the prime(n-1)-smooth numbers, for n>1.
a(n)/A038110(n) is the asymptotic mean of the number of prime(n-1)-smooth divisors of the positive integers, for n>1 (cf. A001511, A072078, A355583). (End)

			A038110(50)/ a(50) = 0.1020..., exp(-gamma)/log(229) = 0.1033...
1*2*4/(2*3*5) = 4/15 has denominator a(4) = 15. - _Jonathan Sondow_, Jan 31 2014

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 429.

Michael De Vlieger, Table of n, a(n) for n = 1..423
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92.
Jonathan Sondow and Eric Weisstein, Euler Product, MathWorld.

Cf. A000203, A002110, A005867, A038110, A038111.
Cf. A236435, A236436.
Cf. A001511, A072078, A355583.

[1] cat [Denominator((&*[NthPrime(k-1)-1:k in [2..n]])/(&*[NthPrime(k-1):k in [2..n]])):n in [2..20]]; // Marius A. Burtea, Sep 19 2019

Table[Denominator@ Product[EulerPhi@ Prime[i]/Prime@ i, {i, n}], {n, 0, 19}] (* Michael De Vlieger, Jan 10 2015 *)
{1}~Join~Denominator@ FoldList[Times, Table[EulerPhi@ Prime[n]/Prime@ n, {n, 19}]] (* Michael De Vlieger, Jul 26 2016 *)
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Denominator@ Table[b[n], {n, 0, 20}] (* Fred Daniel Kline, Jun 27 2017 *)
Join[{1},Denominator[With[{nn=20},FoldList[Times,Prime[Range[nn]]-1]/FoldList[ Times,Prime[Range[nn]]]]]] (* Harvey P. Dale, Apr 17 2022 *)

a(n) = A002110(n) / gcd( A005867(n), A002110(n) ).
A038110(n) / a(n) ~ exp( -gamma ) / log( prime(n) ), Mertens's theorem for x = prime(n) = A000040(n).
A038110(n) / a(n) = A005867(n) / A002110(n). - corrected by Simon Tatham, Jul 26 2016
a(n) = A038111(n) / prime(n). - Vladimir Shevelev, Jan 10 2014
a(n) = A038110(n) + A161527(n-1). - Jamie Morken, Jun 19 2019

Definition corrected by Jonathan Sondow, Jan 31 2014

A084968 Multiples of 7 coprime to 30.

Original entry on oeis.org

7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511, 539, 553, 581, 623, 637, 679, 707, 721, 749, 763, 791, 833, 847, 889, 917, 931, 959, 973, 1001, 1043, 1057, 1099, 1127, 1141, 1169, 1183, 1211, 1253, 1267, 1309

Offset: 1

Robert G. Wilson v, Jun 15 2003

Numbers 7*k such that gcd(k,30) = 1.

Numbers congruent to 7, 49, 77, 91, 119, 133, 161, 203 modulo 210. - Jianing Song, Nov 18 2022

			77 is in the sequence because gcd(77, 30) = 1.
84 is not in the sequence because gcd(84, 3) = 6.
91 is in the sequence because gcd(91, 30) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Subsequence of A008589.
Fourth row of A083140.
Cf. A084967 (5), A084969 (11), A084970 (13), A332799 (17), A332798 (19), A332797 (23), A007775 (7-rough numbers).

q:= k-> igcd(k, 30)=1:
select(q, [7*i$i=1..300])[];  # Alois P. Heinz, Feb 25 2020

7Select[ Range[190], GCD[ #, 2*3*5] == 1 & ]

is(n)=gcd(210,n)==7 \\ Charles R Greathouse IV, Aug 05 2013

G.f.: 7*x*(x^8 + 6*x^7 + 4*x^6 + 2*x^5 + 4*x^4 + 2*x^3 + 4*x^2 + 6*x + 1) / ((x-1)^2*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Feb 24 2013
Lim_{n->oo} a(n)/n = A038111(4)/A038110(4) = 105/4. - Vladimir Shevelev, Jan 20 2015
a(n) = 7*A007775(n).
a(n+8) = a(n) + 210. - Jianing Song, Nov 18 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(23 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/105. - Amiram Eldar, Jul 15 2023

A084970 Numbers whose smallest prime factor is 13.

Original entry on oeis.org

13, 169, 221, 247, 299, 377, 403, 481, 533, 559, 611, 689, 767, 793, 871, 923, 949, 1027, 1079, 1157, 1261, 1313, 1339, 1391, 1417, 1469, 1651, 1703, 1781, 1807, 1937, 1963, 2041, 2119, 2171, 2197, 2249, 2327, 2353, 2483, 2509, 2561, 2587, 2743, 2873

Offset: 1

Robert G. Wilson v, Jun 15 2003

			a(2) = 13*13, a(3) = 13*17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 481.

Sixth row of A083140.

Cf. A084967 (5), A084968 (7), A084969 (11), A332799 (17), A332798 (19), A332797 (23), A008365 (13-rough numbers).

Select[ 13Range@ 225, GCD[#, 2310] == 1 &] (* Robert G. Wilson v, Dec 18 2014 *)

is(n)=n%13==0 && gcd(n,2310)==1 \\ Charles R Greathouse IV, Nov 19 2014

a(n) = a(n-480) + 30030 = a(n-1) + a(n-480) - a(n-481). - Charles R Greathouse IV, Nov 19 2014
Lim_{n->infinity} a(n)/n = A038111(6)/A038110(6) = 1001/16 = 62.5625. - Vladimir Shevelev, Jan 20 2015
a(n) = 13*A008365(n).

More terms from David Wasserman, Oct 19 2004

cp's OEIS Frontend

A161527 Numerators of cumulative sums of rational sequence A038110(k)/A038111(k).

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A307388 Length of the period of decimal representation of Product_{k=1..n} A038111(k)/A038110(k).

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A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).

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A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

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A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

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A038110 Numerator of frequency of integers with smallest divisor prime(n).

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A084967 Multiples of 5 whose GCD with 6 is 1.

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A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).