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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
Programs
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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 *) -
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
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
Extensions
More terms from Vladeta Jovovic, May 15 2003
Comments