A342175 - OEIS

%I #38 Jul 07 2022 02:12:12

%S 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,

%T 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,

%U 1,1,1,1,1,1,19,1,1,11,13,1,1,5,7,1,1,3,1

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

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

%C The conjecture above is false (see A353203 for counterexamples). - _Ivan N. Ianakiev_, Jul 04 2022

%F a(n) = A113496(n) - A002808(n). - _Jon E. Schoenfield_, Mar 04 2021

%e The first composite number is 4, and the smallest larger composite to which it is coprime is 9, so a(1) = 9 - 4 = 5.

%e The second composite number is 6, and the smallest larger composite to which it is coprime is 25, so a(2) = 25 - 6 = 19.

%t 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 *)

%o (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

%o (Python)

%o from sympy import isprime, gcd, composite

%o def A342175(n):

%o     m = composite(n)

%o     k = m+1

%o     while gcd(k,m) != 1 or isprime(k):

%o         k += 1

%o     return k-m # _Chai Wah Wu_, Mar 28 2021

%Y Cf. A002808, A113496, A353203.

%K nonn

%O 1,1

%A _William C. Laursen_, Mar 04 2021