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

1,1

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

Links

Table of n, a(n) for n=1..83.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002808, A113496, A353203.

Sequence in context: A347670 A139237 A317541 * A258080 A318181 A274995

Adjacent sequences: A342172 A342173 A342174 * A342176 A342177 A342178

Keyword

nonn

Author

William C. Laursen, Mar 04 2021

Status

approved