A038388 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.

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

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A038387, A038389.

Cf. A053626, A053627, A253905.

Sequence in context: A366383 A366382 A138221 * A366392 A272573 A328477

Adjacent sequences: A038385 A038386 A038387 * A038389 A038390 A038391

Keyword

nonn,easy,mult

Author

N. J. A. Sloane.

Extensions

More terms from Vladeta Jovovic, May 15 2003

Status

approved