This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A175200 #48 Mar 19 2024 09:34:36
%S A175200 1,6,24,28,40,54,96,120,135,216,224,234,270,360,384,486,496,540,588,
%T A175200 600,640,672,864,891,936,1000,1080,1350,1372,1521,1536,1638,1782,1792,
%U A175200 1920,1944,2016,2160,2176,3000,3240,3375,3402,3456,3564,3724,3744,3780,4320
%N A175200 Numbers k such that rad(k) divides sigma(k).
%C A175200 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 A175200 Also numbers k such that k divides sigma(k)^tau(k). - _Arkadiusz Wesolowski_, Nov 09 2013
%C A175200 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 A175200 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 A175200 Pollack and Pomerance call these numbers "prime-abundant numbers". - _Amiram Eldar_, Jun 02 2020
%D A175200 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 A175200 Donovan Johnson, <a href="/A175200/b175200.txt">Table of n, a(n) for n = 1..10000</a>
%H A175200 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 A175200 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 A175200 rad(6) = 6, sigma(6) = 12 = 6*2.
%e A175200 rad(24) = 6, sigma(24) = 60 = 6*10.
%e A175200 rad(43875) = 195, sigma(43875) = 87360 = 195*448.
%p A175200 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 A175200 Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* _Vincenzo Librandi_, Aug 07 2018 *)
%o A175200 (PARI) isok(n) = {fs = Set(factor(sigma(n))[,1]); fn = Set(factor(n)[,1]); fn == setintersect(fn, fs);} \\ _Michel Marcus_, Nov 03 2015
%o A175200 (Magma) [n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)];// _Vincenzo Librandi_, Aug 07 2018
%Y A175200 Cf. A000203, A007947, A027598, A069235, A105402, A173615, A336554 (odd terms).
%K A175200 nonn
%O A175200 1,2
%A A175200 _Michel Lagneau_, Mar 03 2010