A173190 Values of n such that tau(n) = rad(n)^2, where tau(n) is the number of divisor of n, and rad(n) is the product of the distinct prime factors of n (rad(1)=1).

1, 8, 6561, 6912, 7776, 18432, 52488, 393216, 708588, 258280326, 327680000, 1000000000, 2097152000, 1007769600000, 1612431360000, 1813985280000, 2149908480000, 3936600000000, 6122200320000, 6561000000000, 7652750400000

Offset

1,2

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.1. E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.

Links

Table of n, a(n) for n=1..21.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

C. K. Caldwell, The Prime Glossa, Number of divisors

J. J. Holt & J. W. Jones, Discovering Number Theory, Section 1.4, Counting Divisors

Example

tau(1) = 1, rad(1) = 1, and tau(1) = rad(1)^2 tau(8) = 4, rad(8) = 2, and tau(8) = rad(8)^2 tau(6561) = 9, rad(6561) = 3, and tau(6561) = rad(6561)^2

Maple

with(numtheory): for n from 1 to 50000000 do : t1 := ifactors(n)[2] : t2 := mul(t1[i][1], i=1..nops(t1)): if tau(n) = t2*t2 then print (n): else fi : od :

Crossrefs

Sequence in context: A278384 A114133 A221233 * A291879 A281450 A354564

Adjacent sequences: A173187 A173188 A173189 * A173191 A173192 A173193

Keyword

nonn

Author

Michel Lagneau, Feb 12 2010

Extensions

a(10)-a(21) from Donovan Johnson, Feb 13 2010

Status

approved