A135718 a(n) = smallest divisor of n^2 that is not a divisor of n.
4, 9, 8, 25, 4, 49, 16, 27, 4, 121, 8, 169, 4, 9, 32, 289, 4, 361, 8, 9, 4, 529, 9, 125, 4, 81, 8, 841, 4, 961, 64, 9, 4, 25, 8, 1369, 4, 9, 16, 1681, 4, 1849, 8, 25, 4, 2209, 9, 343, 4, 9, 8, 2809, 4, 25, 16, 9, 4, 3481, 8, 3721, 4, 27, 128, 25, 4, 4489, 8, 9, 4, 5041
Offset: 2
Examples
The divisors of 12 are 1,2,3,4,6,12. The divisors of 12^2 = 144 are 1,2,3,4,6,8,9,12,16,18,24,36,48,72,144. So the smallest divisor of 144 that is not a divisor of 12 is 8.
Links
- Vincenzo Librandi and David A. Corneth, Table of n, a(n) for n = 2..10001 (first 1999 terms from Vincenzo Librandi)
Programs
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Maple
with(numtheory): a:=proc(n) options operator, arrow: op(1, `minus`(divisors(n^2), divisors(n))) end proc: seq(a(n),n=2..60); # Emeric Deutsch, May 18 2008
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Mathematica
a135718[n_] := Map[First[Complement[Divisors[#^2], Divisors[#]]]&, Range[2, n]] a135718[60] (* data *) (* Hartmut F. W. Hoft, Jun 13 2017 *) Table[Min@ Map[Apply[Power, # + {0, 1}] &, FactorInteger@ n], {n, 2, 60}] (* Michael De Vlieger, Jun 23 2017 *)
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PARI
a(n) = fordiv(n^2, x, if (n % x, return (x))); \\ Michel Marcus, Jun 13 2017
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PARI
a(n) = my(f=factor(n)); vecmin(vector(#f~, i, f[i,1]^(f[i,2]+1))) \\ David A. Corneth, Jun 28 2017
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PARI
first(n) = {n++; my(v = vector(n-1), l = List()); forprime(p = 2, n, v[p-1] = p^2); forprime(p = 2, sqrtint(n), pp = p; j = 1; while(pp
Formula
If n = product{p=primes, p|n} p^b(n,p), where each b(n,p) is a positive integer, then a(n) = the minimum value of a p^(b(n,p)+1) where p is a prime that divides n. Example: 24 has the prime factorization of 2^3 *3^1. So a(24) = the minimum of 2^(3+1) and 3^(1+1) = the minimum of 16 and 9, which is 9.
a(p) = p^2 for p prime. - Michel Marcus, Jun 13 2017
Extensions
More terms from Emeric Deutsch, May 18 2008
Comments