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%I M2050 #89 Jul 17 2023 15:07:13
%S 1,2,12,360,75600,174636000,5244319080000,2677277333530800000,
%T 25968760179275365452000000,5793445238736255798985527240000000,
%U 37481813439427687898244906452608585200000000,7517370874372838151564668004911177464757864076000000000,55784440720968513813368002533861454979548176771615744085560000000000
%N Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).
%C Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).
%C Superprimorials, from primorials by analogy with superfactorials.
%C Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.
%C Subsequence of A130091. - _Reinhard Zumkeller_, May 06 2007
%C Hankel transform of A171448. - _Paul Barry_, Dec 09 2009
%C 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 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 Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.
%H T. D. Noe, <a href="/A006939/b006939.txt">Table of n, a(n) for n=0..25</a>
%F 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 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 log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - _Charles R Greathouse IV_, Mar 14 2011
%F 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 a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
%e a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.
%p 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 Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* _Harvey P. Dale_, Jul 07 2011 *)
%t Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* _Alonso del Arte_, Sep 30 2011 *)
%o (PARI) a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ _Charles R Greathouse IV_, Jul 25 2011
%o (Haskell)
%o a006939 n = a006939_list !! n
%o a006939_list = scanl1 (*) a002110_list -- _Reinhard Zumkeller_, Jul 21 2012
%o (Magma) [1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // _G. C. Greubel_, Oct 14 2018
%Y Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).
%Y Cf. A002110, A051357.
%Y A000142 counts divisors of superprimorials.
%Y A000325 counts uniform divisors of superprimorials.
%Y A008302 counts divisors of superprimorials by bigomega.
%Y A022915 counts permutations of prime indices of superprimorials.
%Y A076954 is a sister-sequence.
%Y A118914 has row a(n) equal to {1..n}.
%Y A124010 has row a(n) equal to {n..1}.
%Y A130091 lists numbers with distinct prime multiplicities.
%Y A317829 counts factorizations of superprimorials.
%Y A336417 counts perfect-power divisors of superprimorials.
%Y A336426 gives non-products of superprimorials.
%Y Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.
%K easy,nonn,nice
%O 0,2
%A _N. J. A. Sloane_
%E Corrected and extended by _Labos Elemer_, May 30 2001
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