A381805 Smallest composite squarefree number that is coprime to n.

6, 15, 10, 15, 6, 35, 6, 15, 10, 21, 6, 35, 6, 15, 14, 15, 6, 35, 6, 21, 10, 15, 6, 35, 6, 15, 10, 15, 6, 77, 6, 15, 10, 15, 6, 35, 6, 15, 10, 21, 6, 55, 6, 15, 14, 15, 6, 35, 6, 21, 10, 15, 6, 35, 6, 15, 10, 15, 6, 77, 6, 15, 10, 15, 6, 35, 6, 15, 10, 33, 6, 35

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*q.

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) * A380539(n) = A382248(n)/A020639(n).

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

n < a(n) for n in A051250, a finite sequence whose largest term is 60.

Example

a(1) = 6 = 2*3, since p = 2, q = 3.
a(2) = 15 = 3*5, since p = 3, q = 5.
a(3) = 10 = 2*5, since p = 2, q = 5.
a(4) = 15 = 3*5, since p = 3, q = 5, a(2^i) = 15 for i > 0.
a(6) = 35 = 5*7, since p = 5, q = 7.
a(9) = 20 = 2*5, since p = 2, q = 5, a(3^i) = 10 for i > 0.
a(10) = 21 = 3*7, since p = 3, q = 7.
a(12) = 35 = 5*7, since p = 5, q = 7, a(k) = 35 for n in A033845 (i.e., n such that rad(n) = 6).
a(20) = 21 = 3*7, since p = 3, q = 7, a(k) = 21 for n in A033846 (i.e., n such that rad(n) = 10).
a(30) = 77 = 7*11, 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]; q = NextPrime[q]; c++] ][[-1, 1]], {n, 120}]

Prog

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

Crossrefs

Cf. A002110, A007947, A051250, A053669, A120944, A380539, A382248.

Sequence in context: A202749 A123623 A240990 * A389358 A215739 A387364

Adjacent sequences: A381802 A381803 A381804 * A381806 A381807 A381808

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 31 2025

Status

approved