A104350 Partial products of largest prime factors of numbers <= n.

1, 2, 6, 12, 60, 180, 1260, 2520, 7560, 37800, 415800, 1247400, 16216200, 113513400, 567567000, 1135134000, 19297278000, 57891834000, 1099944846000, 5499724230000, 38498069610000, 423478765710000, 9740011611330000

Offset: 1

Author

Reinhard Zumkeller, Mar 06 2005

Keywords

• nonn

Comments

Partial Products of A006530: a(n) = Product_{k=1..n} A006530(k).

a(n) = a(n-1)*A006530(n) for n>1, a(1) = 1;

A020639(a(n)) = A040000(n-1), A006530(a(n)) = A007917(n) for n>1.

A001221(a(n)) = A000720(n), A001222(a(n)) = A001477(n-1).

A007947(a(n)) = A034386(n).

a(n) = A000142(n) / A076928(n). [Corrected by Franklin T. Adams-Watters, Oct 30 2006]

In decimal representation: A104351(n) = number of digits of a(n), A104355(n) = number of trailing zeros of a(n).

A104357(n) = a(n) - 1, A104365(n) = a(n) + 1.

References

• Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Publ. Inst. Elie Cartan, Vol. 13, Nancy, 1990.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..641
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint, arXiv:1202.3670 [math.HO], 2012-2018.
• Eric Weisstein's World of Mathematics, Greatest Prime Factor.
• Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Crossrefs

Cf. A000142, A002110, A006530, A007947, A020639, A046670, A072486, A076928, A104351, A104355, A104357, A104365.

Cf. A001620, A084945.

Programs

Haskell

a104350 n = a104350_list !! (n-1)
a104350_list = scanl1 (*) a006530_list
-- Reinhard Zumkeller, Apr 10 2014

Mathematica

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; Table[A104350[n], {n, 30}] (* G. C. Greubel, May 09 2017 *)
FoldList[Times,Table[FactorInteger[n][[-1,1]],{n,30}]] (* Harvey P. Dale, May 25 2023 *)

PARI

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=prod(i=2,n,gpf(i)) \\ Charles R Greathouse IV, Apr 29 2015

PARI

first(n)=my(v=vector(n,i,1)); forfactored(k=2,n, v[k[1]]=v[k[1]-1]*vecmax(k[2][,1])); v \\ Charles R Greathouse IV, May 10 2017

Formula

log(a(n)) = c * n * log(n) + c * (1-gamma) * n + O(n * exp(-log(n)^(3/8-eps))), where c is the Golomb-Dickman constant (A084945) and gamma is Euler's constant (A001620) (Tenenbaum, 1990). - Amiram Eldar, May 21 2021

Extensions

More terms from David Wasserman, Apr 24 2008