A058250 -id:A058250 - cp's OEIS frontend

A071349 Numbers k for which the GCD of the k-th primorial number and its totient (A058250) sets record.

1, 2, 4, 5, 9, 10, 15, 16, 17, 23, 27, 28, 35, 39, 40, 41, 43, 49, 56, 57, 61, 62, 64, 66, 69, 72, 73, 76, 77, 91, 92, 96, 97, 102, 103, 104, 107, 111, 114, 117, 119, 127, 128, 137, 139, 143, 146, 150, 154, 155, 166, 171, 181, 182, 186, 195, 196, 201, 208, 214, 215

Offset: 1

Labos Elemer, May 21 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A058250, A002110, A000010, A005867, A071350.

q[n_] := Product[Prime[i], {i, 1, n}]; fq[n_] := Product[Prime[i] - 1, {i, 1, n}];
a=0; Do[s=GCD[q[n], fq[n]]; If[s>a, a=s; Print[n]], {n, 1, 1000}]

If A058250(m) > A058250(k) for all k < m then m is a term.

A071350 Distinct values of A058250; these terms appear first at subscripts listed in A071349.

Original entry on oeis.org

1, 2, 6, 30, 330, 2310, 53130, 690690, 20030010, 821230410, 13960916970, 739928599410, 27377358178170, 2272320728788110, 97709791337888730, 8696171429072096970, 165227257152369842430, 18670680058217792194590

Offset: 1

Labos Elemer, May 21 2002

Michael De Vlieger, Table of n, a(n) for n = 1..341

Cf. A058250, A002110, A000010, A005867, A071349, A112037.

Prepend[FoldList[Times,DeleteDuplicates[Rest[Flatten[FactorInteger[#][[All, 1]]&/@(Prime[Range[100]]-1)]]]],1] (* Jamie Morken, Apr 27 2021 after Harvey P. Dale at A112037, May 26 2019 *)

f(n) = my(pr=prod(k=1, n, prime(k))); gcd(pr, eulerphi(pr)); \\ A058250
lista(nn) = Set(vector(nn, k, f(k))); \\ Michel Marcus, Apr 27 2021

a(n) = a(n-1) * A112037(n), n >= 2. - David A. Corneth, Apr 27 2021

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

A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

Original entry on oeis.org

1, 1, 5, 1, 7, 11, 13, 17, 19, 23, 29, 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209

Offset: 1

Jamie Morken and Michael De Vlieger, May 16 2017

Values in row n of a(n) are those of row n of A286942 complement those of row n of A279864.
From Michael De Vlieger, May 18 2017: (Start)
Numbers t < p_n# such that gcd(t, p_n#) = 0, where p_n# = A002110(n).
Numbers in the reduced residue system of A002110(n).
A005867(n) = number of terms of a(n) in row n; local minimum of Euler's totient function.
A048862(n) = number of primes in row n of a(n).
A048863(n) = number of nonprimes in row n of a(n).
Since 1 is coprime to all n, it delimits the rows of a(n).
The prime A000040(n+1) is the second term in row n since it is the smallest prime coprime to A002110(n) by definition of primorial.
The smallest composite in row n is A001248(n+1) = A000040(n+1)^2.
The Kummer numbers A057588(n) = A002110(n) - 1 are the last terms of rows n, since (n - 1) is less than and coprime to all positive n. (End)

			The triangle starts
1;
1, 5;
1, 7, 11, 13, 17, 19, 23, 29;
1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209;

Michael De Vlieger, Table of n, a(n) for n = 1..6299 (rows 1 <= n <= 6 flattened).
Mathoverflow, Lower bound for Euler's totient for almost all integers. - _Michael De Vlieger_, May 18 2017
Eric Weisstein's World of Mathematics, Totative. - _Michael De Vlieger_, May 18 2017

Cf. A002110, A005867, A048862, A057588, A279864, A286941, A286942, A309497, A038110, A058250, A329815.

Cf. A285784 (nonprimes that appear), A335334 (row sums).

Table[Function[P, Select[Range@ P, CoprimeQ[#, P] &]]@ Product[Prime@ i, {i, n}], {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)

row(n) = my(P=factorback(primes(n))); select(x->(gcd(x, P) == 1), [1..P]); \\ Michel Marcus, Jun 02 2020

More terms from Michael De Vlieger, May 18 2017

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).

Original entry on oeis.org

0, 1, 2, 1, 11, 2, 1, 8, 7, 14, 13, 4, 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8

Offset: 0

Jamie Morken, Aug 05 2019

The sequence is Primorial rows of A308121.
Row n has length A005867(n).
Row n > 1 average value = A060753(n)/2.
Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
First value on row(n) = A161527(n-1).
Last value on row(n) = A038110(n) for n > 2.
For n > 1, A060753(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A060753(n), where a = A005867(n-1) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
Related identity:
  A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

			The triangle starts:
row1: 0;
row2: 1;
row3: 2, 1;
row4: 11, 2, 1, 8, 7, 14, 13, 4;
row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;

Michael De Vlieger, Table of n, a(n) for n = 0..6299 (rows n = 0..6, flattened)
Eric Weisstein's World of Mathematics, Totative

Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.

row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* Amiram Eldar, Aug 10 2019 *)

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

Original entry on oeis.org

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

A335334 Sum of the integers in the reduced residue system of A002110(n).

Original entry on oeis.org

1, 6, 120, 5040, 554400, 86486400, 23524300800, 8045310873600, 4070927302041600, 3305592969257779200, 3074201461409734656000, 4094836346597766561792000, 6715531608420337161338880000, 12128250084807128913378017280000

Offset: 1

Jamie Morken, Jun 02 2020

nonn

Sum of the integers up to A002110(n) and coprime to A002110(n).

The sequence gives the sum of row n of A286941(n).

			For n = 3: A002110(3) = 30, the reduced residue system of 30 is {1, 7, 11, 13, 17, 19, 23, 29}. The sum is a(3) = 120.

Michael De Vlieger, Table of n, a(n) for n = 1..196
Eric Weisstein's World of Mathematics, Totative.

Cf. A023896, A002110, A005867, A070826, A038110, A058250, A286941.

n = 15;
A002110 = Drop[FoldList[Times, 1, Prime[Range[n]]], 1];
A005867 = Drop[EulerPhi@FoldList[Times, 1, Prime@Range@n], 1];
A002110*A005867/2
(* Second program: *)
Map[# EulerPhi[#]/2 &, FoldList[Times, Prime@ Range@ 14]] (* Michael De Vlieger, Apr 07 2021 *)

a(n) = my(P=factorback(primes(n))); P*eulerphi(P)/2; \\ Michel Marcus, Jun 02 2020

a(n) = A023896(A002110(n)).
a(n) = A002110(n)*A005867(n)/2 = A070826(n)*A005867(n).
a(n) = (A002110(n)*A038110(n+1)/2)*A058250(n).

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]

cp's OEIS Frontend

A071349 Numbers k for which the GCD of the k-th primorial number and its totient (A058250) sets record.

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A071350 Distinct values of A058250; these terms appear first at subscripts listed in A071349.

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

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A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)*k-A038110(n)*A286941(n,k).

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

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A335334 Sum of the integers in the reduced residue system of A002110(n).

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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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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).