A072905 a(n) is the least k > n such that k*n is a square.

4, 8, 12, 9, 20, 24, 28, 18, 16, 40, 44, 27, 52, 56, 60, 25, 68, 32, 76, 45, 84, 88, 92, 54, 36, 104, 48, 63, 116, 120, 124, 50, 132, 136, 140, 49, 148, 152, 156, 90, 164, 168, 172, 99, 80, 184, 188, 75, 64, 72, 204, 117, 212, 96, 220, 126, 228, 232, 236, 135, 244, 248

Offset: 1

Author

Benoit Cloitre, Aug 10 2002

Keywords

• easy
• nonn

Comments

From Peter Kagey, Jun 22 2015: (Start)

a(n) is a bijection from the positive integers to A013929 (numbers that are not squarefree). Proof:

(1) Injection: Suppose that b

(2) Surjection: Given some number k in A013929, a(A007913(k)*(A000188(k)-1)^2.) = k (End)

Examples

			12 is the smallest integer > 3 such that 3*12 = 6^2 is a perfect square, hence a(3) = 12.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000188, A007913, A010052, A067722.

Cf. A182448, A253905.

Programs

Haskell

a072905 n = head [k | k <- [n + 1 ..], a010052 (k * n) == 1]
-- Reinhard Zumkeller, Feb 07 2015

Maple

f:= proc(n) local F,f,x,y;
     F:= ifactors(n)[2];
     x:= mul(`if`(f[2]::odd,f[1],1),f=F);
     y:= mul(f[1]^floor(f[2]/2),f=F);
     x*(y+1)^2
end proc:
map(f, [$1..100]); # Robert Israel, Jun 23 2015

Mathematica

a[n_] := For[k = n+1, True, k++, If[IntegerQ[Sqrt[k*n]], Return[k]]]; Array[a, 100] (* Jean-François Alcover, Jan 26 2018 *)

PARI

a(n)=if(n<0,0,s=n+1; while(issquare(s*n)==0,s++); s)

PARI

a(n)=my(c=core(n)); (sqrtint(n/c)+1)^2*c \\ Charles R Greathouse IV, Jun 23 2015

Ruby

def a(n)
  k = Math.sqrt(n).to_i
  k -= 1 until n % k**2 == 0
  n + 2*n/k + n/(k**2)
end # Peter Kagey, Jul 27 2015

Formula

a(n) = n + A067722(n). - Peter Kagey, Feb 05 2015

a(n) = A007913(n)*(A000188(n)+1)^2. - Peter Kagey, Feb 06 2015

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + 2*zeta(3)/zeta(2) + Pi^2/15 = 3.11949956554216757204... . - Amiram Eldar, Feb 17 2024

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