A340735 a(n) is the smallest positive integer that begins a run of exactly 2*n-1 consecutive integers having at least 4 divisors each.

6, 14, 32, 90, 140, 200, 294, 1832, 1070, 888, 1130, 2180, 2478, 2972, 4298, 5592, 1328, 9552, 30594, 19334, 16142, 15684, 81464, 28230, 31908, 19610, 35618, 82074, 44294, 43332, 34062, 89690, 162144, 134514, 173360, 31398, 404598, 212702, 188030, 542604, 265622

Offset: 1

Author

Jon E. Schoenfield, Jan 17 2021

Keywords

• nonn

Comments

If "integers having at least 4 divisors each" in this sequence's definition were replaced with "integers having at least 3 divisors each" (i.e., composite numbers), the resulting sequence would be A045881.

A045881(n) = a(n) except when the run of 2*n-1 consecutive composite numbers beginning with A045881(n) includes a number with exactly 3 divisors (i.e., the square of a prime). The first six such exceptions are as follows:

.

n A045881(n) a(n) 3-divisor number

-- ---------- ---- ----------------

1 4 6 4 = 2^2

2 8 14 9 = 3^2

3 24 32 25 = 5^2

7 114 294 121 = 11^2

9 524 1070 529 = 23^2

12 1670 2180 1681 = 41^2

.

There are no other exceptions among the first 672 terms of A045881 (see the b-file there). Can it be proved that there are no other exceptions?

Examples

			a(1)=6 because 6=2*3 (which has 4 divisors, {1,2,3,6}) is the first isolated number that has at least 4 divisors.
a(2)=14 because 14 is the first number that begins a run of exactly 2*2-1=3 consecutive integers having at least 4 divisors each: tau(14)=tau(2*7)=4; tau(15)=tau(3*5)=4; tau(16)=tau(2^4)=5.
a(3)=32 because 32 is the first number that begins a run of exactly 2*3-1=5 consecutive integers having at least 4 divisors each: tau(32)=tau(2^5)=6; tau(33)=tau(3*11)=4; tau(34)=tau(2*17)=4; tau(35)=tau(5*7)=4; tau(36)=tau(2^2*3^2)=9.
		

Crossrefs

Cf. A045881.