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A175200
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Numbers k such that rad(k) divides sigma(k).
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12
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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
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OFFSET
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1,2
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COMMENTS
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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, ...
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
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REFERENCES
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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.
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LINKS
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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.
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EXAMPLE
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rad(6) = 6, sigma(6) = 12 = 6*2.
rad(24) = 6, sigma(24) = 60 = 6*10.
rad(43875) = 195, sigma(43875) = 87360 = 195*448.
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MAPLE
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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 :
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MATHEMATICA
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Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* Vincenzo Librandi, Aug 07 2018 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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