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

2,1

Comments

a(1) doesn't exist because 1 is coprime to all integers.

Terms are composite since primes either divide or are coprime to other numbers. - Michael De Vlieger, Feb 20 2025

Links

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

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, a(n) = min{p_i^(e_i + 1); i = 1.. ..A001221(n), 2*q}, where p_i^e_i is the greatest power of prime p_i dividing n, p_1 = 2, q = A053669(n); For odd n, a(n) = 2*A020639(n); See Example. - David James Sycamore, Feb 28 2025

a(n) = min(A096014(n), A135718(n)). - Michael De Vlieger, Feb 24 2025

Example

From David James Sycamore, Feb 28 2025: (Start) Using above formula: 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 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)

Mathematica

Table[k = 3; Until[1 < GCD[k, n] < k, k++]; k, {n, 2, 120}] (* Michael De Vlieger, Feb 20 2025 *)

Prog

(PARI) a(n)=for(k=4, 2*n, if(gcd(n, k)>1 && n%k, return(k))) \\ Charles R Greathouse IV, Apr 05 2013
(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

Crossrefs

Cf. A001221, A020639, A053669, A096014, A100484, A133995, A135718.

Sequence in context: A199172 A323674 A183986 * A215454 A155750 A159475

Adjacent sequences: A160992 A160993 A160994 * A160996 A160997 A160998

Keyword

nonn,changed

Author

Leroy Quet, Jun 01 2009

Extensions

Extended by Ray Chandler, Jun 13 2009

Status

approved