A055768 -id:A055768 - cp's OEIS frontend

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

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

A112037 Go through all of the primes p and for each one, factor p-1 into primes. List the primes in order of their first appearance in the p-1 factorizations.

Original entry on oeis.org

2, 3, 5, 11, 7, 23, 13, 29, 41, 17, 53, 37, 83, 43, 89, 19, 113, 131, 67, 47, 73, 31, 79, 173, 179, 61, 191, 97, 233, 239, 251, 127, 139, 281, 71, 293, 101, 103, 107, 163, 59, 359, 193, 199, 137, 419, 431, 443, 151, 491, 509, 181, 109, 277, 593, 149, 307, 641, 653

Offset: 2

Michel Dauchez (mdzdm(AT)yahoo.fr), Nov 28 2005

The length of this list of distinct prime factors of p-1 encountered through p=prime(n) is given by A055768. - Ray Chandler, Nov 30 2005
If "p-1" is changed to "p+1" we get A236388. - N. J. A. Sloane, Jan 24 2014
A permutation of the primes by Dirichlet's theorem on arithmetic progressions: for any pair (r,s) of integers such that gcd(r,s)=1 there are infinitely many primes in the sequence r + k*s; choose r=1 and s=p. - Joerg Arndt, Mar 20 2016

			We start with the second prime, 3. 3-1 = 2, so 2 is the first term.
5-1 = 2*2, nothing new.
7-1 = 2*3 and 3 is new, so that is the second term.
11-1 = 2*5 and we get 5; etc.

Alois P. Heinz, Table of n, a(n) for n = 2..10000
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A055768, A112038, A114461, A236388.

Set(Flat(List(Filtered([3..1500],IsPrime),i->Factors(i-1)))); # Muniru A Asiru, Dec 06 2018

lst = {}; r[n_] := (len = Length@lst; lst = Flatten@ Join[lst, Select[First /@ FactorInteger[Prime@n - 1], ! MemberQ[lst, # ] &]]; If[l < Length@lst, 1, 0]); Do[ r[n], {n, 214}]; lst (* Robert G. Wilson v, Nov 30 2005 *)
DeleteDuplicates[Rest[Flatten[FactorInteger[#][[All,1]]&/@ (Prime[ Range[ 250]]-1)]]] (* Harvey P. Dale, May 26 2019 *)

g=1;forprime(p=2,299,f=factorint(p-1)[,1];z=factorback(f); r=z/gcd(z,g);g*=r;if(r>1,print(r," ",p))); \\ Jack Brennen, Nov 28 2005

Better description from Jack Brennen, Nov 28 2005

Extended by Ray Chandler and Robert G. Wilson v, Nov 30 2005

A055769 Largest prime dividing phi of the n-th primorial.

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 5, 5, 11, 11, 11, 11, 11, 11, 23, 23, 29, 29, 29, 29, 29, 29, 41, 41, 41, 41, 41, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 83, 83, 89, 89, 89, 89, 89, 89, 89, 89, 113, 113, 113, 113, 113, 113, 113, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131

Offset: 1

Labos Elemer, Jul 12 2000

nonn

			While the largest prime factors of 10th, 100th or 1000th primorials are 29, 541, 7919, those of their totients are 11, 251, 3911, respectively.

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

Cf. A000010, A002110, A005867, A055768.

a055769 = a006530 . a005867  -- Reinhard Zumkeller, May 01 2013

Map[FactorInteger[EulerPhi@ #][[-1, 1]] &, FoldList[#1 #2 &, Prime@ Range@ 66]] (* Michael De Vlieger, Oct 26 2017 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=my(p=prime(n),q=1); while(2*q+1Charles R Greathouse IV, Dec 03 2014

a(n) = A006530(A000010(A002110(n))). [corrected by Amiram Eldar, Sep 18 2024]

a(n) = A006530(A005867(n)). - Reinhard Zumkeller, May 01 2013

A382789 The number of prime factors of Euler phi of the n-th primorial number, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 3, 5, 7, 10, 14, 17, 19, 22, 25, 29, 33, 36, 38, 41, 43, 47, 50, 53, 58, 61, 63, 67, 73, 77, 80, 82, 87, 92, 96, 99, 103, 106, 109, 113, 117, 122, 124, 127, 129, 134, 137, 144, 148, 152, 156, 159, 161, 165, 169, 172, 178, 182, 190, 192, 195, 200, 204

Offset: 0

Amiram Eldar, Apr 05 2025

Amiram Eldar, Table of n, a(n) for n = 0..10000

Partial sums of A023508.

Cf. A000010, A002110, A001222, A055768, A055769.

Join[{0}, Accumulate[PrimeOmega[Prime[Range[100]] - 1]]]

list(nmax) = {my(s = 0, c = 0); print1(s, ", "); forprime(p = 1, , c++; s += bigomega(p-1); print1(s, ", "); if(c == nmax, break));}

a(n) = A001222(A000010(A002110(n))).
a(n) = A001222(A005867(n)).
a(n) = Sum_{k=1..n} A023508(k).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A112037 Go through all of the primes p and for each one, factor p-1 into primes. List the primes in order of their first appearance in the p-1 factorizations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Extensions

A055769 Largest prime dividing phi of the n-th primorial.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A382789 The number of prime factors of Euler phi of the n-th primorial number, counted with multiplicity.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula