A175200 Numbers k such that rad(k) divides sigma(k).

1, 6, 24, 28, 40, 54, 96, 120, 135, 216, 224, 234, 270, 360, 384, 486, 496, 540, 588, 600, 640, 672, 864, 891, 936, 1000, 1080, 1350, 1372, 1521, 1536, 1638, 1782, 1792, 1920, 1944, 2016, 2160, 2176, 3000, 3240, 3375, 3402, 3456, 3564, 3724, 3744, 3780, 4320

Offset: 1

Author

Michel Lagneau, Mar 03 2010

Keywords

• nonn

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

Examples

			rad(6) = 6, sigma(6) = 12 = 6*2.
rad(24) = 6, sigma(24) = 60 = 6*10.
rad(43875) = 195, sigma(43875) = 87360 = 195*448.
		

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.

Crossrefs

Cf. A000203, A007947, A027598, A069235, A105402, A173615, A336554 (odd terms).

Programs

Magma

[n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)];// Vincenzo Librandi, Aug 07 2018

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 *)

PARI

isok(n) = {fs = Set(factor(sigma(n))[,1]); fn = Set(factor(n)[,1]); fn == setintersect(fn, fs);} \\ Michel Marcus, Nov 03 2015