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.

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

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