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

1,2

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.

Links

Table of n, a(n) for n=1..38.

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A138858 A076967 A233247 * A231180 A250029 A111378

Adjacent sequences: A363654 A363655 A363656 * A363658 A363659 A363660

Keyword

nonn

Author

Simon Jensen, Jun 13 2023

Status

approved