A067722 -id:A067722 - cp's OEIS frontend

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

Benoit Cloitre, Aug 10 2002

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)

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

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

Cf. A000188, A007913, A010052, A067722.

Cf. A182448, A253905.

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

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

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

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

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

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

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

A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

Original entry on oeis.org

1, 2, 8, 1, 64, 256, 2048, 4, 1, 131072, 262144, 32, 8388608, 33554432, 134217728, 1, 2147483648, 8, 34359738368, 1024, 549755813888, 4398046511104, 17592186044416, 8192, 2, 1125899906842624, 32, 65536, 72057594037927936, 576460752303423488

Offset: 1

Peter Kagey, Jun 08 2018

nonn

Conjecture: a(n) > 0.
If the conjecture is true, all terms are powers of two, and a(n) >= A259527(n).
a(n) = 0 if and only if A066400(n) = 2.
a(n) = 0 if and only if A255167(n) = 0.
a(n) <= 2^(A067722(n) - 1). - Peter Kagey, Nov 13 2018

			For n = 3, the a(3) = 8 subsets of {4, 5, ..., 11} with a product with squarefree part of 3 are {4, 5, 6, 9, 10}, {4, 5, 6, 10}, {4, 6, 8}, {4, 6, 8, 9}, {5, 6, 9, 10}, {5, 6, 10}, {6, 8}, and {6, 8, 9}.

Peter Kagey, Table of n, a(n) for n = 1..200

Cf. A005117, A006255, A066400, A072905, A255167.

A321482 a(n) = log_2(A305677(n)).

Original entry on oeis.org

0, 1, 3, 0, 6, 8, 11, 2, 0, 17, 18, 5, 23, 25, 27, 0, 31, 3, 35, 10, 39, 42, 44, 13, 1, 50, 5, 16, 56, 59, 62, 4, 66, 69, 70, 0, 76, 77, 80, 25, 84, 86, 89, 29, 14, 95, 98, 8, 1, 5, 106, 34, 111, 18, 117, 39, 121, 123, 125, 42, 129, 132, 21, 2, 139, 141, 144

Offset: 1

Peter Kagey, Nov 12 2018

nonn

a(n) <= A067722(n) - 1.

It is conjectured that A305677(n) > 0. If A305677(n) = 0, then define a(n) = -1.

Peter Kagey, Table of n, a(n) for n = 1..300

Cf. A067722, A305677.

Showing 1-3 of 3 results.

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

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A305677 Number of subsets of {n+1, n+2, ..., A072905(n)-1} whose product has the same squarefree part as n.

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A321482 a(n) = log_2(A305677(n)).

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