A342175 a(n) is the difference between the n-th composite number and the smallest larger composite to which it is relatively prime.

5, 19, 1, 1, 11, 13, 1, 1, 5, 7, 1, 1, 3, 1, 1, 1, 1, 5, 19, 1, 1, 1, 1, 13, 1, 1, 9, 13, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 17, 1, 1, 1, 1, 19, 1, 1, 11, 5, 1, 1, 1, 1, 7, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 19, 1, 1, 11, 13, 1, 1, 5, 7, 1, 1, 3, 1

Offset: 1

Author

William C. Laursen, Mar 04 2021

Keywords

• nonn

Comments

Conjecture: The only nonprime terms are squares (based on checking the first 2 million terms). - Ivan N. Ianakiev, Mar 28 2021

The conjecture above is false (see A353203 for counterexamples). - Ivan N. Ianakiev, Jul 04 2022

Examples

			The first composite number is 4, and the smallest larger composite to which it is coprime is 9, so a(1) = 9 - 4 = 5.
The second composite number is 6, and the smallest larger composite to which it is coprime is 25, so a(2) = 25 - 6 = 19.
		

Crossrefs

Cf. A002808, A113496, A353203.

Programs

Mathematica

Table[Block[{k = 1}, While[Nand[GCD[#, k] == 1, CompositeQ[# + k]], k++]; k] &@ FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1], {n, 83}] (* Michael De Vlieger, Mar 19 2021 *)

PARI

lista(nn) = {forcomposite(c=1, nn, my(x=c+1); while (isprime(x) || (gcd(x,c) != 1), x++); print1(x - c, ", "););} \\ Michel Marcus, Mar 04 2021

Python

from sympy import isprime, gcd, composite
def A342175(n):
    m = composite(n)
    k = m+1
    while gcd(k,m) != 1 or isprime(k):
        k += 1
    return k-m # Chai Wah Wu, Mar 28 2021

Formula

a(n) = A113496(n) - A002808(n). - Jon E. Schoenfield, Mar 04 2021