A375929 Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.

3, 4, 7, 8, 11, 12, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 76, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94

Offset: 1

Author

Gus Wiseman, Sep 12 2024

Keywords

• nonn

Comments

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

Examples

			The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after positions 3, 4, 7, 8, ...
		

Crossrefs

Positions in A002808 of each element of A068780.

The complement is A065890 shifted.

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 nonprime numbers is A375926.

A000040 lists the prime numbers, differences A001223.

A000961 lists prime-powers (inclusive), differences A057820.

A002808 lists the composite numbers, differences A073783.

A018252 lists the nonprime numbers, differences A065310.

A046933 counts composite numbers between primes.

Cf. A014689, A054546, A176246, A246655, A251092.

Programs

Mathematica

Join@@Position[Differences[Select[Range[100],CompositeQ]],1]

Python

from sympy import primepi
def A375929(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+2+y))-2
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

Python

# faster for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    pic, prevc = 0, -1
    for i in count(4):
        if not isprime(i):
            if i == prevc + 1:
                yield pic
            pic, prevc = pic+1, i
print(list(islice(agen(), 10000))) # Michael S. Branicky, Sep 17 2024

Formula

a(n) = A375926(n) - 1.