OFFSET
1,2
COMMENTS
rad(k) is the product of the distinct primes dividing k (A007947). sigma(k) is the sum of divisors of k (A000203). The odd numbers in this sequence (A336554) are rare: 1, 135, 891, 1521, 3375, 5733, 10935, 11907, 41067, 43875, ...
Also numbers k such that k divides sigma(k)^tau(k). - Arkadiusz Wesolowski, Nov 09 2013
This sequence is infinite. It contains an infinite number of even elements and an infinite number of odd ones. This is due to the fact that for every odd prime p and every prime q dividing p+1, p*q^r is prime-perfect when r = -1 + the multiplicative order of q modulo p. - Emmanuel Vantieghem, Oct 13 2014
For each term, it is possible to find an exponent k such that sigma(n)^k is divisible by n. A007691 (multi-perfect numbers) is a subsequence of terms that have k=1. A263928 is the subsequence of terms that have k=2. - Michel Marcus, Nov 03 2015
Pollack and Pomerance call these numbers "prime-abundant numbers". - Amiram Eldar, Jun 02 2020
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. 827.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..10000
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].
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12A, Paper A14, 2012.
EXAMPLE
rad(6) = 6, sigma(6) = 12 = 6*2.
rad(24) = 6, sigma(24) = 60 = 6*10.
rad(43875) = 195, sigma(43875) = 87360 = 195*448.
MAPLE
for n from 1 to 5000 do : p1:= ifactors(n)[2] :p2 :=mul(p1[i][1], i=1..nops(p1)): if irem(sigma(n), p2) =0 then print (n): else fi: od :
MATHEMATICA
Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* Vincenzo Librandi, Aug 07 2018 *)
PROG
(PARI) isok(n) = {fs = Set(factor(sigma(n))[, 1]); fn = Set(factor(n)[, 1]); fn == setintersect(fn, fs); } \\ Michel Marcus, Nov 03 2015
(Magma) [n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)]; // Vincenzo Librandi, Aug 07 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 03 2010
STATUS
approved