A217056 Highly composite numbers (A002182) which are the product of 4 consecutive integers (A052762).

24, 120, 360, 840, 1680, 5040, 17297280

Offset: 1

Author

Raphie Frank, Sep 25 2012

Keywords

• nonn
• more

Comments

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.

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).

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).

Question: Is this sequence complete?

Next term > A002182(1000) = 3.3826...*10^76. - Joerg Arndt, Oct 07 2012

Examples

			24 = 1*2*3*4 = 5^2 - 1; 24/24 = 1.
120 = 2*3*4*5 = 11^2 - 1; 120/24 = 5.
360 = 3*4*5*6 = 19^2 - 1; 360/24 = 15.
840 = 4*5*6*7 = 29^2 - 1; 840/24 = 35.
1680 = 5*6*7*8 = 41^2 - 1; 1680/24 = 70.
5040 = 7*8*9*10 = 71^2 - 1; 5040/24 = 210.
17297280 = 63*64*65*66 = 4159^2 - 1; 17297280/24 = 720720.
{5, 11, 19, 29, 41, 71, 4159} are all primes one less than a pronic number.
{1, 5, 15, 35, 70, 210, 720720} are all generalized pentagonal numbers.
		

Crossrefs

Cf. A163264, A002182, A072825.

Formula

A217056 = A002182 intersect A052762. - M. F. Hasler, Oct 18 2013