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Let f(n) be the smallest number such that the arithmetic mean (A) and geometric mean (G) of n and f(n) are both integers; sequence gives G values.
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%I #26 Oct 27 2022 07:31:41

%S 1,2,3,4,5,6,7,4,3,10,11,12,13,14,15,8,17,6,19,20,21,22,23,12,5,26,9,

%T 28,29,30,31,8,33,34,35,12,37,38,39,20,41,42,43,44,15,46,47,24,7,10,

%U 51,52,53,18,55,28,57,58,59,60,61,62,21,16,65,66,67,68,69,70,71,12,73,74,15

%N Let f(n) be the smallest number such that the arithmetic mean (A) and geometric mean (G) of n and f(n) are both integers; sequence gives G values.

%H Amiram Eldar, <a href="/A038388/b038388.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^((e+1)/2) if e is odd, 2^(e/2+1) if p=2 and e is even, p^(e/2) if p>2 and e is even. - _Vladeta Jovovic_, May 15 2003

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (13/24)*zeta(3)/zeta(2) = 0.395829... . - _Amiram Eldar_, Oct 27 2022

%t Table[k = 1; While[Nand @@ IntegerQ /@ {(n + k)/2, g = Sqrt[n*k]}, k++]; g, {n, 75}] (* _Jayanta Basu_, Jul 14 2013 *)

%t f[p_, e_] := If[OddQ[e], p^((e + 1)/2), If[p == 2, 2^(e/2 + 1), p^(e/2)]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 09 2020*)

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^((f[i,2]+1)/2), if(f[i,1]==2, 2^(f[i,2]/2+1), f[i,1]^(f[i,2]/2))));} \\ _Amiram Eldar_, Oct 27 2022

%Y Cf. A038387, A038389.

%Y Cf. A053626, A053627, A253905.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_.

%E More terms from _Vladeta Jovovic_, May 15 2003