A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

1, 2, 12, 360, 75600, 174636000, 5244319080000, 2677277333530800000, 25968760179275365452000000, 5793445238736255798985527240000000, 37481813439427687898244906452608585200000000, 7517370874372838151564668004911177464757864076000000000, 55784440720968513813368002533861454979548176771615744085560000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).

Superprimorials, from primorials by analogy with superfactorials.

Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.

Subsequence of A130091. - Reinhard Zumkeller, May 06 2007

Hankel transform of A171448. - Paul Barry, Dec 09 2009

This might be a good place to explain the name "Chernoff sequence" since his name does not appear in the References or Links as of Mar 22 2014. - Jonathan Sondow, Mar 22 2014

Pickover (1992) named this sequence after Paul Chernoff of California, who contributed this sequence to his book. He was possibly referring to American mathematician Paul Robert Chernoff (1942 - 2017), a professor at the University of California. - Amiram Eldar, Jul 27 2020

Examples

			a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.
		

References

• Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.

Links

• T. D. Noe, Table of n, a(n) for n=0..25

Crossrefs

Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).

Cf. A002110, A051357.

A000142 counts divisors of superprimorials.

A000325 counts uniform divisors of superprimorials.

A008302 counts divisors of superprimorials by bigomega.

A022915 counts permutations of prime indices of superprimorials.

A076954 is a sister-sequence.

A118914 has row a(n) equal to {1..n}.

A124010 has row a(n) equal to {n..1}.

A130091 lists numbers with distinct prime multiplicities.

A317829 counts factorizations of superprimorials.

A336417 counts perfect-power divisors of superprimorials.

A336426 gives non-products of superprimorials.

Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.

Programs

Haskell

a006939 n = a006939_list !! n
a006939_list = scanl1 (*) a002110_list -- Reinhard Zumkeller, Jul 21 2012

Magma

[1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // G. C. Greubel, Oct 14 2018

Maple

a := []; printlevel := -1; for k from 0 to 20 do a := [op(a),product(ithprime(i)^(k-i+1),i=1..k)] od; print(a);

Mathematica

Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* Harvey P. Dale, Jul 07 2011 *)
Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* Alonso del Arte, Sep 30 2011 *)

PARI

a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ Charles R Greathouse IV, Jul 25 2011

Python

from math import prod
from sympy import prime
def A006939(n): return prod(prime(k)**(n-k+1) for k in range(1,n+1)) # Chai Wah Wu, Aug 12 2025

Formula

a(n) = m(1)*m(2)*m(3)*...*m(n), where m(n) = n-th primorial number. - N. J. A. Sloane, Feb 20 2005

a(0) = 1, a(n) = a(n - 1)p(n)#, where p(n)# is the n-th primorial A002110(n) (the product of the first n primes). - Alonso del Arte, Sep 30 2011

log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - Charles R Greathouse IV, Mar 14 2011

A181796(a(n)) = A000110(n+1). It would be interesting to have a bijective proof of this theorem, which is stated at A181796 without proof. See also A336420. - Gus Wiseman, Aug 03 2020

Extensions

Corrected and extended by Labos Elemer, May 30 2001