A175200 - OEIS

%I #48 Mar 19 2024 09:34:36

%S 1,6,24,28,40,54,96,120,135,216,224,234,270,360,384,486,496,540,588,

%T 600,640,672,864,891,936,1000,1080,1350,1372,1521,1536,1638,1782,1792,

%U 1920,1944,2016,2160,2176,3000,3240,3375,3402,3456,3564,3724,3744,3780,4320

%N Numbers k such that rad(k) divides sigma(k).

%C 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, ...

%C Also numbers k such that k divides sigma(k)^tau(k). - _Arkadiusz Wesolowski_, Nov 09 2013

%C 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

%C 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

%C Pollack and Pomerance call these numbers "prime-abundant numbers". - _Amiram Eldar_, Jun 02 2020

%D 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.

%H Donovan Johnson, <a href="/A175200/b175200.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Paul Pollack and Carl Pomerance, <a href="http://www.emis.de/journals/INTEGERS/papers/a14self/a14self.Abstract.html">Prime-Perfect Numbers</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12A, Paper A14, 2012.

%e rad(6) = 6, sigma(6) = 12 = 6*2.

%e rad(24) = 6, sigma(24) = 60 = 6*10.

%e rad(43875) = 195, sigma(43875) = 87360 = 195*448.

%p 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 :

%t Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* _Vincenzo Librandi_, Aug 07 2018 *)

%o (PARI) isok(n) = {fs = Set(factor(sigma(n))[,1]); fn = Set(factor(n)[,1]); fn == setintersect(fn, fs);} \\ _Michel Marcus_, Nov 03 2015

%o (Magma) [n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)];// _Vincenzo Librandi_, Aug 07 2018

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

%K nonn

%O 1,2

%A _Michel Lagneau_, Mar 03 2010