A279083 - OEIS

%I #14 Jan 22 2017 21:40:50

%S 1,3,4,6,8,12,16,18,20,24,28,30,32,36,40

%N Numbers k such that there exists at least one tetrahedral number with exactly k divisors.

%C The only odd terms are 1, 3, and 45 (which correspond to the three positive tetrahedral numbers that are square, i.e., 1, 4, and 19600). It is easy to show that no term larger than 6 is semiprime.

%C A tetrahedral number with exactly 42 divisors would have to be of the form p^6 * q^2 * r, with p, q, and r distinct primes; does such a tetrahedral number exist?

%C A tetrahedral number with exactly 50 divisors would have to be of the form p^4 * q^4 * r, with p, q, and r distinct primes; does such a tetrahedral number exist?

%C Additional terms < 200 include (but may not be limited to) 44, 45, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 192, 196

%Y Cf. A000292, A279081, A279082.

%K nonn,more

%O 1,2

%A _Jon E. Schoenfield_, Jan 06 2017