A330737 a(n) is the first index k in A002182 (highly composite numbers) from which onward all terms A002182(i), i >= k, are multiples of the n-th prime, a(0) = 1 by convention.

1, 2, 4, 9, 15, 28, 38, 55, 71, 92, 110, 125, 146, 167, 183, 206, 225, 258, 281, 313, 339, 363, 399, 425, 453, 488, 515, 550, 585, 618, 657, 705, 739, 794, 830, 866, 902, 950, 999, 1036, 1074, 1113, 1151, 1198, 1234, 1270, 1306, 1347, 1393, 1436, 1479, 1528, 1571, 1615, 1671, 1719, 1774, 1824, 1875, 1925, 1975, 2026, 2087, 2170, 2235

Offset: 0

Author

Antti Karttunen, Dec 29 2019

Keywords

• nonn

Comments

Equivalently, a(n) is the first index k in A002182 from which onward all terms A002182(i), i >= k, are multiples of A002110(n), the n-th primorial number.

Question: Is this sequence well-defined for any n > 1? For all n? See also A199337.

Note that this differs from A072846 at n = 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, ...

Yes, the sequence is well defined for all n, see A199337 for proof that all A002182(k) >= A329571(n)^2 are divisible by n. - M. F. Hasler, Jan 07 2020

Examples

			a(0) = 1 as A002110(0) = 1, and A002182(1) = 1, and as all integers are divisible by 1, including all terms of A002182.
A002182(9) = 60, and because from then onward all highly composite numbers are multiples of 30 (= A002110(3) = prime(1)*prime(2)*prime(3)), we have a(3) = 9.
		

Links

• Eric Weisstein's World of Mathematics, Highly Composite Number.
• Wikipedia, Highly composite number.

Crossrefs

Cf. A002110, A002182, A072846, A199337, A328520, A329571.

Programs

PARI

\\ v002182 contains the terms of A002182 up to some suitably big value:
A330737(n) = if(!n,1,my(x=prime(n)); forstep(k=#v002182,1,-1,if(v002182[n]%x,return(1+k))));