A363657 Numbers m where A217854(m) is a record minimum.

1, 4, 9, 16, 36, 100, 144, 324, 400, 576, 900, 1764, 2304, 3600, 7056, 8100, 14400, 28224, 32400, 44100, 57600, 108900, 112896, 129600, 176400, 396900, 435600, 518400, 608400, 705600, 1587600, 2822400, 5336100, 6350400, 14288400, 15681600, 17640000, 21344400

Offset: 1

Author

Simon Jensen, Jun 13 2023

Keywords

• nonn

Comments

(-m)^tau(m) < 0 and (-m)^tau(m) < (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.

All terms are squares.

It is conjectured that if m is a term, then abs((-m)^tau(m)) <= abs((-k)^tau(k)) for some k < m. See the link.

Examples

			9 is a term since (-9)^tau(9) = (-9)^3 = -729 and -729 < (-k)^tau(k) for k = 1,...,8.
25 is not a term since (-25)^tau(5) = (-25)^3 = -15625 > (-16)^tau(16) = (-16)^5 = -1048576 and 16 < 25.
		

Links

• Simon K. Jensen, On an extended divisor product summatory function

Crossrefs

Cf. A363658, A000290, A067128, A000005, A217854, A224914.

Programs

PARI

isok(m) = my(x=(-m)^numdiv(m)); for (k=1, m-1, if (x >= (-k)^numdiv(k), return(0))); return(1); \\ Michel Marcus, Jun 18 2023