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

1,1

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.

Links

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

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A066610 A176696 A269031 * A113164 A237252 A017234

Adjacent sequences: A382764 A382765 A382766 * A382768 A382769 A382770

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 04 2025

Status

approved