A363050 Lesser of two consecutive integers such that one has more prime factors (counted with multiplicity), but the other has more divisors.

495, 728, 729, 975, 1071, 1424, 1616, 1700, 1862, 1875, 2024, 2144, 2223, 2349, 2384, 2415, 2624, 2996, 3104, 3124, 3125, 3159, 3184, 3483, 3663, 4095, 4130, 4292, 4304, 4335, 4779, 4976, 5103, 5312, 5427, 5535, 5589, 5624, 5775, 6224, 6416, 6544, 6560, 6704

Offset

1,1

Comments

The first pair of consecutive integers that are both terms is (728, 729), and the first run of three consecutive integers all appearing as terms begins at 127251 (see Examples). The first run of four consecutive integers appearing as terms begins at 4405832. Do arbitrarily long runs of consecutive integers occur?

Links

Table of n, a(n) for n=1..44.

Example

495 = 3*3*5*11 has 4 prime factors and 12 divisors, while 496 = 2*2*2*2*31 has 5 prime factors but only 10 divisors, so 495 is a term.
728 = 2*2*2*7*13, 729 = 3*3*3*3*3*3, and 730 = 2*5*73 have 5, 6, and 3 prime factors and 16, 7, and 8 divisors, respectively, so both 728 and 729 are terms.
127251 = 3*3*3*3*1571, 127252 = 2*2*29*1097, 127253 = 7*7*7*7*53, and 127254 = 2*3*127*167 have 5, 4, 5, and 4 prime factors, and 10, 12, 10, and 16 divisors, respectively, so 127251, 127252, and 127253 are all terms.

Crossrefs

Cf. A000005, A001222.

Sequence in context: A264327 A045007 A062903 * A347891 A059828 A160851

Adjacent sequences: A363047 A363048 A363049 * A363051 A363052 A363053

Keyword

nonn

Author

Jon E. Schoenfield, May 14 2023

Status

approved