

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.


3



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, 28, 29, 30, 31, 2, 33, 34, 35, 4, 37, 38, 39, 10, 41, 42, 43, 44, 5, 46, 47, 12, 1, 2, 51, 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
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OFFSET

1,2


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)


FORMULA

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)
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
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/80 = 0.3701101... .  Amiram Eldar, Oct 27 2022


MATHEMATICA

Table[k = 1; While[Nand @@ IntegerQ /@ {(n + k)/2, Sqrt[n*k]}, k++]; k, {n, 78}] (* Jayanta Basu, Jul 14 2013 *)
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 *)


PROG

(PARI) a(n)={for(k=1, n, if((n+k)%2==0 && issquare(n*k), return(k)))} \\ Andrew Howroyd, Feb 12 2018
(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


CROSSREFS

Cf. A038388, A038389.
Cf. A053626, A053627.
Sequence in context: A037903 A037841 A165111 * A240832 A173527 A043267
Adjacent sequences: A038384 A038385 A038386 * A038388 A038389 A038390


KEYWORD

nonn,easy,mult


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Vladeta Jovovic, May 15 2003


STATUS

approved



