A375926 Numbers k such that A018252(k+1) = A018252(k) + 1. In other words, the k-th nonprime number is 1 less than the next.

4, 5, 8, 9, 12, 13, 15, 16, 17, 18, 21, 22, 23, 24, 26, 27, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Author

Gus Wiseman, Sep 11 2024

Keywords

• nonn

Examples

			The nonprime numbers are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after term 4, term 5, term 8, etc.
		

Crossrefs

The complement appears to be A014689, except the first term.

Positions of 1's in A065310 (see also A054546, A073783).

First differences are A373403 (except first).

The version for non-prime-powers is A375713, differences A373672.

The version for prime-powers is A375734, differences A373671.

The version for non-perfect-powers is A375740.

The version for composite numbers is A375929.

A000040 lists the prime numbers, differences A001223.

A018252 lists the nonprimes, exclusive A002808.

A046933 counts composite numbers between primes.

Cf. A000961, A006549, A057820, A176246, A246655, A251092, A375708.

Programs

Mathematica

Join@@Position[Differences[Select[Range[100],!PrimeQ[#]&]],1]

Python

from sympy import primepi
def A375926(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+bisection(lambda y:primepi(x+1+y))-1
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024