A168263 For any m < n, and for all values of k, d(n^k) > d(m^k). (Let k, m, and n represent positive integers only.)

1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720

Offset: 1

Author

Matthew Vandermast, Nov 23 2009

Keywords

• fini
• full
• nonn
• changed

Comments

d(n) is the number of divisors of n (A000005(n)).

All members must be highly composite numbers (A002182) with at least as many distinct prime factors as any smaller positive integer (A116998). (See Formula and Example sections.) It turns out that these two conditions are jointly sufficient.

Ramanujan proved that a) for any prime p, there exist a finite number of highly composite numbers with p as its largest prime factor; and b) in the canonical prime factorization of a highly composite number with largest prime factor p, the exponents for all primes > p are never smaller than they are in the factorization of A003418(p). (See formula 54 of the Ramanujan paper.)

It follows that, if the intersection of A003418 and A116998 is finite, so is the intersection of A002182 and A116998. For proof that the former intersection is finite, see A168262.

By using the given formula for the number of divisors, it is possible to define a canonical polynomial p_n(k) for every natural number n. For example, because 60 = (2^2)(3^1)(5^1), we define p_60(k) = (1+2k)(1+k)(1+k). The present sequence is defined only by examining whether p_n(k) achieves a record for natural numbers k, but the question could also be asked whether p_n(k) achieves a record for all k > 0. This stricter requirement does not hold for a(7)-a(13) at various positive values of k < 1, but it does hold for a(1)-a(6). The present sequence is "full", so a(1)-a(6) are the only numbers to satisfy the stronger property. - Hal M. Switkay, Aug 17 2025

Examples

			1) 1680 has more divisors than any smaller positive integer; thus for all m < n, d(1680^1) > d(m^1).
2) Since the exponents in 1680's prime factorization are (4,1,1,1), the k-th power of 1680 has (4k+1)(k+1)^3 = 4k^4 + 13k^3 + 15k^2 + 7k + 1 divisors. Comparison with the analogous formulas for all smaller members of A025487 shows the following:
a) No number smaller than 1680 has a positive coefficient in its "power formula" for any exponent larger than k^4.
b) The only power formula with a k^4 coefficient as high as 4 is that for 1260 (4k^4 + 12k^3 + 13k^2 + 6k + 1).
c) The k^3 coefficient for 1680 is higher than for 1260.
So for all sufficiently high values of k, d(1680^k) > d(m^k) for all m < 1680.
3) Careful comparison of 1680's "power formula" with the analogous formulas for smaller members of A025487 shows that no intermediate value of k can exist for which d(m^k) >= d(1680^k) if m < 1680.
		

References

• S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Links

• Anonymous?, Polynomial Calculator
• S. Ramanujan, Highly Composite Numbers (p. 15) (note especially pp. 11-15)
• G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Crossrefs

Intersection of A002182 and A116998. Also, intersection of A002182 and A060735, and of A002182 and A168264. (A168264 is a subsequence of A060735, which is a subsequence of A116998.)

Formula

If the canonical factorization of n into prime powers is Product p^e(p), then the formula for the number of divisors of the k-th power of n is Product_p (ek + 1). (See also A146289, A146290.)

For two positive integers m and n with different prime signatures, let j be the largest exponent of k for which m and n have different coefficients, after the above formula for each integer is expanded as a polynomial. Let m_j and n_j denote the corresponding coefficients. d(n^k) > d(m^k) for all sufficiently high values of k if and only if n_j > m_j.