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A038387 a(n) is the smallest number such that the arithmetic mean (A) and geometric mean (G) of n and a(n) are both integers.

%I A038387 #32 Oct 27 2022 07:31:45
%S A038387 1,2,3,4,5,6,7,2,1,10,11,12,13,14,15,4,17,2,19,20,21,22,23,6,1,26,3,
%T A038387 28,29,30,31,2,33,34,35,4,37,38,39,10,41,42,43,44,5,46,47,12,1,2,51,
%U A038387 52,53,6,55,14,57,58,59,60,61,62,7,4,65,66,67,68,69,70,71,2,73,74,3,76,77,78
%N A038387 a(n) is the smallest number such that the arithmetic mean (A) and geometric mean (G) of n and a(n) are both integers.
%H A038387 Amiram Eldar, <a href="/A038387/b038387.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)
%F A038387 If n is odd with prime factorization p1^j1*p2^j2*...pk^jk then f(n)=p1^(j1 mod 2)*p2^(j2 mod 2)*... If n is even then f(n)=2^(2-(j1 mod 2))*p2^(j2 mod 2)*... f(n)=n whenever n is squarefree or n/2 is squarefree. (_Christian G. Bower_)
%F A038387 Multiplicative with a(p^e) = p if e is odd, 4 if p=2 and e is even, 1 if p>2 and e is even. - _Vladeta Jovovic_, May 15 2003
%F A038387 Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/80 = 0.3701101... . - _Amiram Eldar_, Oct 27 2022
%t A038387 Table[k = 1; While[Nand @@ IntegerQ /@ {(n + k)/2, Sqrt[n*k]}, k++]; k, {n, 78}] (* _Jayanta Basu_, Jul 14 2013 *)
%t A038387 f[p_, e_] := If[OddQ[e], p, If[p == 2, 4, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 09 2020 *)
%o A038387 (PARI) a(n)={for(k=1, n, if((n+k)%2==0 && issquare(n*k), return(k)))} \\ _Andrew Howroyd_, Feb 12 2018
%o A038387 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1], if(f[i,1]==2, 4, 1)));} \\ _Amiram Eldar_, Oct 27 2022
%Y A038387 Cf. A038388, A038389.
%Y A038387 Cf. A053626, A053627.
%K A038387 nonn,easy,mult
%O A038387 1,2
%A A038387 _N. J. A. Sloane_
%E A038387 More terms from _Vladeta Jovovic_, May 15 2003