A382248 Smallest number k that is neither squarefree nor a prime power such that k is coprime to n.

12, 45, 20, 45, 12, 175, 12, 45, 20, 63, 12, 175, 12, 45, 28, 45, 12, 175, 12, 63, 20, 45, 12, 175, 12, 45, 20, 45, 12, 539, 12, 45, 20, 45, 12, 175, 12, 45, 20, 63, 12, 275, 12, 45, 28, 45, 12, 175, 12, 63, 20, 45, 12, 175, 12, 45, 20, 45, 12, 539, 12, 45, 20

Offset: 1

Author

Michael De Vlieger, Mar 31 2025

Keywords

• nonn
• easy

Comments

Let p be the smallest prime that is coprime to n and let q be the second smallest prime that is coprime to n. Then a(n) = p^2 * q.

Records in this sequence are set by n in A002110.

Examples

			a(1) = 12 = 2^2*3, since p = 2, q = 3.
a(2) = 45 = 3^2*5, since p = 3, q = 5.
a(3) = 20 = 2^2*5, since p = 2, q = 5.
a(4) = 45 = 3^2*5, since p = 3, q = 5, a(2^i) = 45 for i > 0.
a(6) = 175 = 5^2*7, since p = 5, q = 7.
a(9) = 20 = 2^2*5, since p = 2, q = 5, a(3^i) = 20 for i > 0.
a(10) = 63 = 3^2*7, since p = 3, q = 7.
a(12) = 175 = 5^2*7, since p = 5, q = 7, a(k) = 175 for n in A033845 (i.e., n such that rad(n) = 6).
a(20) = 63 = 3^2*7, since p = 3, q = 7, a(k) = 63 for n in A033846 (i.e., n such that rad(n) = 10).
a(30) = 539 = 7^2*11, since p = 7, q = 11, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002110, A007947, A053669, A126706, A380539.

Programs

Mathematica

Table[c = 0; q = 2; Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]

PARI

a(n) = my(k=2); while (isprimepower(k) || issquarefree(k) || (gcd(k, n) != 1) , k++); k; \\ Michel Marcus, Apr 01 2025

Formula

a(n) = A053669(n)^2 * A380539(n).

For k and m such that rad(k) = rad(m), a(k) = a(m), where rad = A007947.