A382767 Smallest number k that is powerful but not a prime power that is also coprime to n.

36, 225, 100, 225, 36, 1225, 36, 225, 100, 441, 36, 1225, 36, 225, 196, 225, 36, 1225, 36, 441, 100, 225, 36, 1225, 36, 225, 100, 225, 36, 5929, 36, 225, 100, 225, 36, 1225, 36, 225, 100, 441, 36, 3025, 36, 225, 196, 225, 36, 1225, 36, 441, 100, 225, 36, 1225

Offset: 1

Author

Michael De Vlieger, Apr 04 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^2.

Records in this sequence are set by n in A002110.

Examples

			a(1) = 36 = (2*3)^2, since p = 2, q = 3.
a(2) = 225 = (3*5)^2, since p = 3, q = 5.
a(3) = 100 = (2*5)^2, since p = 2, q = 5.
a(4) = 225 = (3*5)^2, since p = 3, q = 5, a(2^i) = 225 for i > 0.
a(6) = 1225 = (5*7)^2, since p = 5, q = 7.
a(9) = 400 = (2*5)^2, since p = 2, q = 5, a(3^i) = 100 for i > 0.
a(10) = 441 = (3*7)^2, since p = 3, q = 7.
a(12) = 1225 = (5*7)^2, since p = 5, q = 7, a(k) = 1225 for n in A033845 (i.e., n such that rad(n) = 6), where rad = A007947.
a(20) = 441 = (3*7)^2, since p = 3, q = 7, a(k) = 441 for n in A033846 (i.e., n such that rad(n) = 10).
a(30) = 5929 = (7*11)^2, since p = 7, q = 11, etc.
		

Links

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

Crossrefs

Cf. A002110, A007947, A053669, A286708, A380539, A381805, A382248.

Programs

Mathematica

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

Formula

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

a(n) = A381805(n)^2.

a(n) = (A382248(n)/A020639(n))^2.

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