A173320 Numbers k such that tau(sigma(k)) = sopf(k).

2, 3, 4, 15, 16, 42, 45, 64, 81, 84, 245, 336, 340, 342, 460, 539, 550, 580, 605, 684, 882, 1012, 1014, 1160, 1344, 1360, 1640, 1674, 1700, 1785, 1840, 1972, 2178, 2254, 2320, 2322, 2736, 3096, 3348, 3645, 4048, 4096, 4212, 4332, 4389, 4400, 4644, 4830, 5022

Offset: 1

Author

Michel Lagneau, Feb 16 2010

Keywords

• nonn

Comments

sopf(k) is the sum of the distinct primes dividing k (A008472), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of divisor of k (A000203).

Examples

			sigma(2) = 3, tau(3) = 2 and sopf(2) = 2 sigma(2254) = 4104, tau(4104) = 32 and sopf(2254) = 32.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• C. K. Caldwell, The Prime Glossary, Number of divisors
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
• Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Crossrefs

Cf. A000005, A000203, A001414 (sopfr), A001222.

Programs

Maple

with(numtheory): for n from 1 to 12000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)): if tau(sigma(n)) = t2 then print (n): else fi : od :

Mathematica

Select[Range[2,5100],DivisorSigma[0,DivisorSigma[1,#]]==Total[ FactorInteger[ #][[All,1]]]&] (* Harvey P. Dale, May 31 2019 *)

Formula

k such that A062068(k)= A008472(k).