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

%I A006939 M2050 #96 Aug 12 2025 02:39:45
%S A006939 1,2,12,360,75600,174636000,5244319080000,2677277333530800000,
%T A006939 25968760179275365452000000,5793445238736255798985527240000000,
%U A006939 37481813439427687898244906452608585200000000,7517370874372838151564668004911177464757864076000000000,55784440720968513813368002533861454979548176771615744085560000000000
%N A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).
%C A006939 Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).
%C A006939 Superprimorials, from primorials by analogy with superfactorials.
%C A006939 Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.
%C A006939 Subsequence of A130091. - _Reinhard Zumkeller_, May 06 2007
%C A006939 Hankel transform of A171448. - _Paul Barry_, Dec 09 2009
%C A006939 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
%C A006939 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
%D A006939 Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
%D A006939 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006939 James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.
%H A006939 T. D. Noe, <a href="/A006939/b006939.txt">Table of n, a(n) for n=0..25</a>
%F A006939 a(n) = m(1)*m(2)*m(3)*...*m(n), where m(n) = n-th primorial number. - _N. J. A. Sloane_, Feb 20 2005
%F A006939 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
%F A006939 log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - _Charles R Greathouse IV_, Mar 14 2011
%F A006939 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
%e A006939 a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
%e A006939 a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.
%p A006939 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);
%t A006939 Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* _Harvey P. Dale_, Jul 07 2011 *)
%t A006939 Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* _Alonso del Arte_, Sep 30 2011 *)
%o A006939 (PARI) a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ _Charles R Greathouse IV_, Jul 25 2011
%o A006939 (Haskell)
%o A006939 a006939 n = a006939_list !! n
%o A006939 a006939_list = scanl1 (*) a002110_list -- _Reinhard Zumkeller_, Jul 21 2012
%o A006939 (Magma) [1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // _G. C. Greubel_, Oct 14 2018
%o A006939 (Python)
%o A006939 from math import prod
%o A006939 from sympy import prime
%o A006939 def A006939(n): return prod(prime(k)**(n-k+1) for k in range(1,n+1)) # _Chai Wah Wu_, Aug 12 2025
%Y A006939 Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).
%Y A006939 Cf. A002110, A051357.
%Y A006939 A000142 counts divisors of superprimorials.
%Y A006939 A000325 counts uniform divisors of superprimorials.
%Y A006939 A008302 counts divisors of superprimorials by bigomega.
%Y A006939 A022915 counts permutations of prime indices of superprimorials.
%Y A006939 A076954 is a sister-sequence.
%Y A006939 A118914 has row a(n) equal to {1..n}.
%Y A006939 A124010 has row a(n) equal to {n..1}.
%Y A006939 A130091 lists numbers with distinct prime multiplicities.
%Y A006939 A317829 counts factorizations of superprimorials.
%Y A006939 A336417 counts perfect-power divisors of superprimorials.
%Y A006939 A336426 gives non-products of superprimorials.
%Y A006939 Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.
%K A006939 easy,nonn,nice
%O A006939 0,2
%A A006939 _N. J. A. Sloane_
%E A006939 Corrected and extended by _Labos Elemer_, May 30 2001