A341781 Refactorable numbers (or tau numbers, A033950) k such that k/tau(k) is even, where tau(k) = A000005(k).

8, 12, 36, 72, 80, 96, 128, 180, 240, 252, 288, 384, 396, 448, 468, 480, 560, 612, 640, 672, 684, 720, 828, 864, 880, 896, 972, 1040, 1044, 1056, 1116, 1152, 1200, 1248, 1332, 1344, 1360, 1408, 1440, 1476, 1520, 1548, 1620, 1632, 1664, 1680, 1692, 1800, 1824

Offset: 1

Author

Amiram Eldar, Feb 19 2021

Keywords

• nonn

Comments

Zelinsky (2002) called these numbers p-generators. He proved that these are the tau numbers k such that for any prime p, if p does not divide k then p*k is also a tau number. He used these numbers to prove that the number of tau numbers not exceeding m is > pi(m)/2 for all m > 7.42*10^13, where pi(m) = A000720(m).

Examples

			8 is a term since 8/tau(8) = 8/4 = 2 is even.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Crossrefs

Cf. A000005, A000720, A033950, A036762, A039819, A281188.

Programs

Mathematica

q[n_] := Divisible[n, (d = DivisorSigma[0, n])] && EvenQ[n/d]; Select[Range[2000], q]

PARI

isok(k) = my(q=k/numdiv(k)); (denominator(q)==1) && ((q%2) == 0); \\ Michel Marcus, Feb 20 2021