cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A058254 a(n) = lcm{prime(i)-1, i=1..n}.

Original entry on oeis.org

1, 1, 2, 4, 12, 60, 60, 240, 720, 7920, 55440, 55440, 55440, 55440, 55440, 1275120, 16576560, 480720240, 480720240, 480720240, 480720240, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400, 532748591575200
Offset: 0

Views

Author

Labos Elemer, Dec 06 2000

Keywords

Comments

A002110(n) divides b^(a(n)+1) - b for every integer b. - Thomas Ordowski, Nov 24 2014
What is the asymptotic growth of this sequence? a(n) <= A005867(n) <= A002110(n) < e^((1 + o(1))n log n) but this is a large overestimate. - Charles R Greathouse IV, Dec 03 2014
Alexander Kalmynin gives a proof that log a(n) = O(p log log p/log p) where p is the n-th prime, see the MathOverflow link. - Charles R Greathouse IV, Sep 17 2021

Examples

			For n = 5 and 6: a(5) = a(6) = LCM[1, 2, 4, 6, 10, 12] = 60.

Links

Crossrefs

Cf. A000010, A000142, A002110, A003418, A005867, A006093, A055769, A058255.

Programs

  • Haskell
    a058254 n = a058254_list !! (n-1)
a058254_list = scanl1 lcm a006093_list
-- Reinhard Zumkeller, May 01 2013
  • Maple
    seq(ilcm(seq(ithprime(i)-1,i=1..n)), n=0..100); # Robert Israel, Nov 24 2014
  • Mathematica
    Table[LCM @@ (Prime@ Range[1, n] - 1), {n, 27}] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    a(n)=lcm(apply(p->p-1, primes(n))) \\ Charles R Greathouse IV, Dec 03 2014

Formula

a(n) = A002322(A002110(n)). - Thomas Ordowski, Nov 24 2014

Extensions

Offset corrected by Reinhard Zumkeller, May 01 2013
a(0)=1 prepended by Alois P. Heinz, Apr 01 2021

A055768 Number of distinct primes dividing phi of n-th primorial number.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 27, 28, 28
Offset: 1

Views

Author

Labos Elemer, Jul 12 2000

Keywords

Examples

			For primorials with 10, 100, or 1000 prime factors, their totients have only 5, 32 or 241 prime divisors, corresponding to a(10), a(100), and a(1000).

Links

Crossrefs

Cf. A002110, A000010, A001221, A055769.

Programs

  • Haskell
    a055768 = a001221 . a005867  -- Reinhard Zumkeller, May 01 2013
  • Mathematica
    Table[PrimeNu@ EulerPhi[Product[Prime@ i, {i, n}]], {n, 78}] (* or *)
With[{nn = 78}, PrimeNu@ FoldList[LCM @@ {#1, #2} &, Prime@ Range@ nn - 1]] (* Michael De Vlieger, Jul 14 2017 *)
  • PARI
    a(n)=omega(lcm(apply(p->p-1, primes(n)))) \\ Charles R Greathouse IV, Sep 02 2015

Formula

a(n) = A001221(A000010(A002110(n))) = A001221(A005867(n)).
a(n) < n. - Charles R Greathouse IV, Sep 02 2015

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

Views

Author

Amiram Eldar, Apr 05 2025

Keywords

Links

Crossrefs

Partial sums of A023508.
Cf. A000010, A002110, A001222, A055768, A055769.

Programs

  • Mathematica
    Join[{0}, Accumulate[PrimeOmega[Prime[Range[100]] - 1]]]
  • PARI
    list(nmax) = {my(s = 0, c = 0); print1(s, ", "); forprime(p = 1, , c++; s += bigomega(p-1); print1(s, ", "); if(c == nmax, break));}

Formula

a(n) = A001222(A000010(A002110(n))).
a(n) = A001222(A005867(n)).
a(n) = Sum_{k=1..n} A023508(k).
Showing 1-3 of 3 results.

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