A173615 Numbers n such that rad(n)^2 divides sigma(n).

1, 96, 864, 1080, 1782, 6144, 7128, 7776, 17280, 27000, 28512, 54432, 55296, 69984, 87480, 114048, 215622, 276480, 381024, 393216, 432000, 433026, 456192, 497664, 629856, 675000, 862488, 1382400, 1399680, 1677312, 1732104, 1824768, 2187000, 2195424, 2667168

Offset

1,2

Comments

rad(n) is the product of the primes dividing n (A007947) and sigma(n) = sum of divisors of n (A000203). Considering the integers n = (2^a)*(3^b), where a+1 = 6k and b >= 1, we obtain an infinite number of numbers such that rad(n)^2 divides sigma(n).

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.

Links

Donovan Johnson, Table of n, a(n) for n = 1..300

K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12. See Final remarks pp. 10-11.

W. Sierpinski, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Example

rad(96)^2 = 6^2 = 36, sigma(96) = 252 and 36 divides 252

Maple

for n from 1 to 2000000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if irem(sigma(n), t2^2) = 0 then print (n): else fi: od :

Prog

(PARI) isok(n) = my(f=factor(n)); (sigma(f) % factorback(f[, 1])^2) == 0; \\ Michel Marcus, Nov 09 2020

Crossrefs

Cf. A000203, A007947.

Sequence in context: A253410 A326576 A202890 * A103846 A203979 A296960

Adjacent sequences: A173612 A173613 A173614 * A173616 A173617 A173618

Keyword

nonn

Author

Michel Lagneau, Feb 22 2010

Extensions

a(30)-a(35) from Donovan Johnson, Jan 14 2012

Status

approved