%I
%S 24,120,360,840,1680,5040,17297280
%N Highly composite numbers (A002182) which are the product of 4 consecutive integers (A052762).
%C All elements of this sequence, a subset of A163264, are 24 times an element of binomial(n,4) (A000332) and are, therefore, also 24 times a generalized pentagonal number (A001318) since all elements of binomial(n,4) are generalized pentagonal.
%C Additionally, sqrt(a(n) + 1) is prime for these 7 terms. It follows that, at least to a(7), the sum of the divisors of sqrt(a(n) + 1) is a pronic number (A002378).
%C 48 = 2*(1*2*3*4)= 7^2  1 is the only known highly composite number one less than a square that is not a part of this sequence. 48 is also 24 times a generalized pentagonal number and one less than the square of a prime (see also A072825).
%C Question: Is this sequence complete?
%C Next term > A002182(1000) = 3.3826...*10^76.  _Joerg Arndt_, Oct 07 2012
%F A217056 = A002182 intersect A052762.  _M. F. Hasler_, Oct 18 2013
%e 24 = 1*2*3*4 = 5^2  1; 24/24 = 1.
%e 120 = 2*3*4*5 = 11^2  1; 120/24 = 5.
%e 360 = 3*4*5*6 = 19^2  1; 360/24 = 15.
%e 840 = 4*5*6*7 = 29^2  1; 840/24 = 35.
%e 1680 = 5*6*7*8 = 41^2  1; 1680/24 = 70.
%e 5040 = 7*8*9*10 = 71^2  1; 5040/24 = 210.
%e 17297280 = 63*64*65*66 = 4159^2  1; 17297280/24 = 720720.
%e {5, 11, 19, 29, 41, 71, 4159} are all primes one less than a pronic number.
%e {1, 5, 15, 35, 70, 210, 720720} are all generalized pentagonal numbers.
%Y Cf. A163264, A002182, A072825.
%K nonn,more
%O 1,1
%A _Raphie Frank_, Sep 25 2012
