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A175200 Numbers k such that rad(k) divides sigma(k). 8
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 (list; graph; refs; listen; history; text; internal format)
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 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 :

# Alternative

select(n->sigma(n)^n mod n=0, [$1..4320]); # Paolo P. Lava, Aug 07 2018

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

Cf. A027598, A069235, A105402, A173615.

Sequence in context: A072710 A273124 A069235 * A293453 A118372 A263928

Adjacent sequences:  A175197 A175198 A175199 * A175201 A175202 A175203

KEYWORD

nonn

AUTHOR

Michel Lagneau, Mar 03 2010

STATUS

approved

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Last modified July 5 20:21 EDT 2020. Contains 335473 sequences. (Running on oeis4.)