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A066862 Numbers k such that k divides Sum_{i=1..k} gcd(k,i) = A018804(k).

%I A066862 #51 Apr 09 2022 06:16:05
%S A066862 1,4,15,16,27,48,60,64,108,144,240,256,325,432,729,891,960,1008,1024,
%T A066862 1200,1280,1296,1300,1728,1875,2916,3072,3125,3564,3645,3840,3888,
%U A066862 4095,4096,5200,6000,6237,6375,6400,6912,7056,7500,8775,9216,11520,11664,12500
%N A066862 Numbers k such that k divides Sum_{i=1..k} gcd(k,i) = A018804(k).
%C A066862 Also k such that Sum_{d|k} phi(d)/d is an integer. - _Benoit Cloitre_, Apr 14 2002
%C A066862 If two coprime numbers are terms then their product is as well, because Pillai's function A018804(n) is multiplicative. - _Thomas Ordowski_, Oct 28 2014
%C A066862 The first six squarefree terms are 1, 15=3*5, 1488251=19*29*37*73, 4464753=3*19*29*37*73, 7441255=5*19*29*37*73 and 22323765=3*5*19*29*37*73. Are there any others? - _Michel Marcus_ and _Thomas Ordowski_, Nov 01 2014
%H A066862 Charles R Greathouse IV, <a href="/A066862/b066862.txt">Table of n, a(n) for n = 1..3287</a>
%H A066862 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html">A survey of gcd-sum functions</a>, J. Int. Seq., Vol. 13 (2010), Article 10.8.1.
%F A066862 If n = 4^k with k >= 0, n is in the sequence.
%F A066862 If p is prime and k >= 0 then n = p^(kp) is in the sequence. - _Thomas Ordowski_, Oct 28 2014
%p A066862 A066862:=n->`if`(add(gcd(n,i), i=1..n) mod n = 0, n, NULL):
%p A066862 seq(A066862(n), n=1..500); # _Wesley Ivan Hurt_, Oct 28 2014
%t A066862 a066862[n_Integer] := Select[Range[n], Divisible[Sum[GCD[#, i], {i, 1, #}], #] &]; a066862[12500] (* _Michael De Vlieger_, Nov 23 2014 *)
%t A066862 f[p_, e_] := (e*(p - 1)/p + 1); r[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[12500], IntegerQ[r[#]] &] (* _Amiram Eldar_, Apr 09 2022 *)
%o A066862 (PARI) isok(n) = sum(i=1,n,gcd(n,i)) % n == 0; \\ _Michel Marcus_, Nov 20 2013
%o A066862 (PARI) A018804(n)=my(f=factor(n)); prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2])
%o A066862 is(n)=A018804(n)%n==0 \\ _Charles R Greathouse IV_, Oct 28 2014
%Y A066862 Cf. A018804.
%K A066862 nonn
%O A066862 1,2
%A A066862 _Benoit Cloitre_, Jan 25 2002
%E A066862 More terms from _Michel Marcus_, Nov 20 2013