A160995 The smallest positive integer neither a divisor of n nor coprime to n.
4, 6, 6, 10, 4, 14, 6, 6, 4, 22, 8, 26, 4, 6, 6, 34, 4, 38, 6, 6, 4, 46, 9, 10, 4, 6, 6, 58, 4, 62, 6, 6, 4, 10, 8, 74, 4, 6, 6, 82, 4, 86, 6, 6, 4, 94, 9, 14, 4, 6, 6, 106, 4, 10, 6, 6, 4, 118, 8, 122, 4, 6, 6, 10, 4, 134, 6, 6, 4, 142, 10, 146, 4, 6, 6, 14, 4, 158, 6, 6, 4, 166, 8, 10, 4, 6
Offset: 2
Keywords
Examples
From _David James Sycamore_, Feb 28 2025: (Start)
Using my formula above: n = 4235 = 5*7*11^2, so a(n) = 2*5 = 10.
For n = odd prime p, a(n) = 2*p.
For n = 2, a(n) = min{2^2, 2*3} = 4.
For n = 4, a(n) = min{2^3, 2*3} = 6. (For all n = 2^k, k >= 2, a(n) = 6.)
For n = 120 = 2^3*3*5, a(n) = min{16, 9, 25, 14} = 9.
For n = 5040 = 2^4*3^2*5*7, a(n) = min{32, 27, 25, 49, 22} = 22.
For n = 3603600 = 2^4*3^2*5^2*7*11*13, a(n) = min{32,27,125,49,121,169,34} = 27. (End)
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
Programs
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Mathematica
Table[k = 3; Until[1 < GCD[k, n] < k, k++]; k, {n, 2, 120}] (* Michael De Vlieger, Feb 20 2025 *) -
PARI
a(n)=for(k=4,2*n,if(gcd(n,k)>1 && n%k, return(k))) \\ Charles R Greathouse IV, Apr 05 2013
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PARI
a(n)=my(f=factor(n),b);forprime(p=2,,if(n%p,b=p*f[1,1];break));for(i=1,#f[,1],b=min(b,f[i,1]^(f[i,2]+1)));b \\ Charles R Greathouse IV, Apr 05 2013
Formula
For composite n > 4, a(n) is the first term of row n of A133995. - Michael De Vlieger, Feb 20 2025
For even n whose prime factorization is Product_{i=1..k} (p_i)^(e_i), a(n) = min({p_i^(e_i + 1) : i = 1..k} U {2*q}), where q = A053669(n); for odd n, a(n) = 2*A020639(n); see Example. - David James Sycamore, Feb 28 2025 [edited by Peter Munn, Jul 20 2025]
Extensions
Extended by Ray Chandler, Jun 13 2009
Comments