A069235 -id:A069235 - cp's OEIS frontend

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

Michel Lagneau, Mar 03 2010

nonn

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

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

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.

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.

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

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

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 :

Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* Vincenzo Librandi, Aug 07 2018 *)

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

A274205 Numbers such that the sum of divisors is twice the sum of the exponential divisors.

Original entry on oeis.org

6, 24, 54, 216, 1638, 6552, 14256, 55860, 80262, 276822, 321048, 502740, 1107288, 1396500, 1724976, 12568500, 13564278, 20165460, 54257112, 168836850, 181489140, 504136500, 675347400, 4537228500, 28533427650, 60950102850, 114133710600, 162252212850, 243800411400, 649008851400, 734916514878

Offset: 1

Paolo P. Lava, Jun 13 2016

nonn

All terms appear to be multiples of 6.
Subset of A011775, A069235, A175200, A215142.
a(32) > 10^12. If p*r is a term, where p is prime and r is not divisible by p, then p^3*r is also a term. - Giovanni Resta, Jun 15 2016

			Divisors of 6 are 1, 2, 3 and 6 which sum to 12. The only exponential divisor is 6. Finally 12 / 6 = 2.
Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 which sum to 60. Exponential divisors are 6, 24 and their sum is 30. Finally 60 / 30 = 2.

Cf. A000203, A011775, A051377, A069235, A175200, A215142.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n,ok;
for n from 2 to q do a:=ifactors(n)[2]; b:=sort([op(divisors(n))]); c:=0;
for k from 2 to nops(b) do d:=ifactors(b[k])[2]; if nops(d)=nops(a) then
ok:=1; for j from 1 to nops(d) do if not type(a[j][2]/d[j][2],integer) then ok:=0; break; fi; od;
if ok=1 then c:=c+b[k]; fi;  fi; od; if sigma(n)=2*c  then print(n); fi; od; end: P(10^9);

Select[Range[10^6], 2 Times @@ Map[Sum[First[#]^d, {d, Divisors@ Last@ #}] &, FactorInteger@ #] == DivisorSigma[1, #] &] (* Michael De Vlieger, Jun 16 2016 *)

a(16)-a(31) from Giovanni Resta, Jun 15 2016

cp's OEIS Frontend

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

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